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1 {-# OPTIONS --allow-unsolved-metas #-}
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2
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431
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3 open import Level
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4 open import Ordinals
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5 module cardinal {n : Level } (O : Ordinals {n}) where
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6
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7 open import zf
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8 open import logic
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1095
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9 -- import OD
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10 import OD hiding ( _⊆_ )
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11
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431
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12 import ODC
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13 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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431
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14 open import Relation.Binary.PropositionalEquality
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15 open import Data.Nat.Properties
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431
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16 open import Data.Empty
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17 open import Relation.Nullary
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18 open import Relation.Binary
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19 open import Relation.Binary.Core
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20
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21 open inOrdinal O
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22 open OD O
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23 open OD.OD
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24 open ODAxiom odAxiom
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25 open import ZProduct O
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26
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27 import OrdUtil
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28 import ODUtil
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29 open Ordinals.Ordinals O
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30 open Ordinals.IsOrdinals isOrdinal
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31 open Ordinals.IsNext isNext
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32 open OrdUtil O
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33 open ODUtil O
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34
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35 _⊆_ : ( A B : HOD ) → Set n
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36 _⊆_ A B = {x : Ordinal } → odef A x → odef B x
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37
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38
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39 open _∧_
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40 open _∨_
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41 open Bool
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42 open _==_
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43
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44 open HOD
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45
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46 -- _⊗_ : (A B : HOD) → HOD
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47 -- A ⊗ B = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) ))
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48
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49 record Func (A B : HOD) : Set n where
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50 field
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51 func : Ordinal → Ordinal
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52 is-func : {x : Ordinal } → odef A x → odef B (func x)
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53
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54 data FuncHOD (A B : HOD) : (x : Ordinal) → Set n where
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55 felm : (F : Func A B) → FuncHOD A B (& ( Replace A ( λ x → < x , (* (Func.func F (& x))) > )))
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56
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57 FuncHOD→F : {A B : HOD} {x : Ordinal} → FuncHOD A B x → Func A B
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58 FuncHOD→F {A} {B} (felm F) = F
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59
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60 FuncHOD=R : {A B : HOD} {x : Ordinal} → (fc : FuncHOD A B x) → (* x) ≡ Replace A ( λ x → < x , (* (Func.func (FuncHOD→F fc) (& x))) > )
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61 FuncHOD=R {A} {B} (felm F) = *iso
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62
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63 --
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64 -- Set of All function from A to B
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65 --
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66 Funcs : (A B : HOD) → HOD
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67 Funcs A B = record { od = record { def = λ x → FuncHOD A B x } ; odmax = osuc (& (A ⊗ B))
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68 ; <odmax = λ {y} px → subst ( λ k → k o≤ (& (A ⊗ B)) ) &iso (⊆→o≤ (lemma1 px)) } where
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69 lemma1 : {y : Ordinal } → FuncHOD A B y → {x : Ordinal} → odef (* y) x → odef (A ⊗ B) x
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70 lemma1 {y} (felm F) {x} yx with subst (λ k → odef k x) *iso yx
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71 ... | record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef (A ⊗ B) k ) (sym x=ψz) (
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72 product→ (subst (λ k → odef A k) (sym &iso) az) (subst (λ k → odef B k)
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73 (trans (cong (Func.func F) (sym &iso)) (sym &iso)) (Func.is-func F az) ))
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74
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75 record Injection (A B : Ordinal ) : Set n where
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76 field
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77 i→ : (x : Ordinal ) → odef (* A) x → Ordinal
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78 iB : (x : Ordinal ) → ( lt : odef (* A) x ) → odef (* B) ( i→ x lt )
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79 iiso : (x y : Ordinal ) → ( ltx : odef (* A) x ) ( lty : odef (* A) y ) → i→ x ltx ≡ i→ y lty → x ≡ y
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80
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81 record OrdBijection (A B : Ordinal ) : Set n where
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82 field
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83 fun← : (x : Ordinal ) → odef (* A) x → Ordinal
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84 fun→ : (x : Ordinal ) → odef (* B) x → Ordinal
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85 funB : (x : Ordinal ) → ( lt : odef (* A) x ) → odef (* B) ( fun← x lt )
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86 funA : (x : Ordinal ) → ( lt : odef (* B) x ) → odef (* A) ( fun→ x lt )
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87 fiso← : (x : Ordinal ) → ( lt : odef (* B) x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x
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88 fiso→ : (x : Ordinal ) → ( lt : odef (* A) x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x
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89
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90 ordbij-refl : { a b : Ordinal } → a ≡ b → OrdBijection a b
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91 ordbij-refl {a} refl = record {
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92 fun← = λ x _ → x
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93 ; fun→ = λ x _ → x
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94 ; funB = λ x lt → lt
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95 ; funA = λ x lt → lt
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96 ; fiso← = λ x lt → refl
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97 ; fiso→ = λ x lt → refl
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98 }
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99
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100 open Injection
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101 open OrdBijection
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102
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103 record IsImage (a b : Ordinal) (iab : Injection a b ) (x : Ordinal ) : Set n where
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104 field
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105 ax : odef (* a) x
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106 bx : odef (* b) (i→ iab _ ax)
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107
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108 Image : { a b : Ordinal } → Injection a b → HOD
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109 Image {a} {b} iab = record { od = record { def = λ x → IsImage a b iab x } ; odmax = a ; <odmax = λ lt → ? }
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110
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111 image=a : ?
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112 image=a = ?
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113
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114 _=c=_ : ( A B : HOD ) → Set n
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115 A =c= B = OrdBijection ( & A ) ( & B )
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116
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117 c=→≡ : {A B : HOD} → A =c= B → (A ≡ ?) ∧ (B ≡ ?)
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118 c=→≡ = ?
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119
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120 ≡→c= : {A B : HOD} → A ≡ B → A =c= B
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121 ≡→c= = ?
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122
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123 open import BAlgebra O
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124
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125 _-_ : (a b : Ordinal ) → Ordinal
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126 a - b = & ( (* a) ＼ (* b) )
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127
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128 -→< : (a b : Ordinal ) → (a - b) o≤ a
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129 -→< a b = ?
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130
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131 Bernstein1 : {a b : Ordinal } → a o< b → Injection a b ∧ Injection b a → Injection (b - a) b ∧ Injection b (b - a)
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132 Bernstein1 = ?
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133
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134 Bernstein : {a b : Ordinal } → Injection a b → Injection b a → OrdBijection a b
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135 Bernstein {a} {b} iab iba = be00 where
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136 a⊆b : * a ⊆ * b
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137 a⊆b {x} ax = subst (λ k → odef (* b) k) be01 ( iB iab _ ax ) where
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138 be01 : i→ iab x ax ≡ x
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139 be01 = ?
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140 be02 : x ≡ i→ iba x ?
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141 be02 = iiso iab ? ? ax ( iB iba _ ? ) ?
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142 b⊆a : * b ⊆ * a
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143 b⊆a bx = ?
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144 be05 : {a b : Ordinal } → a o< b → Injection a b → Injection b a → ⊥
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145 be05 {a} {b} a<b iab iba = TransFinite0 {λ x → (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥ }
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146 ind a b a<b iab iba where
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147 ind :(x : Ordinal) →
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148 ((y : Ordinal) → y o< x → (b : Ordinal) → y o< b → Injection y b → Injection b y → ⊥ ) →
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149 (b : Ordinal) → x o< b → Injection x b → Injection b x → ⊥
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150 ind x prev b x<b ixb ibx = ?
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151 be00 : OrdBijection a b
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152 be00 with trio< a b
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153 ... | tri< a ¬b ¬c = ⊥-elim ( be05 a iab iba )
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154 ... | tri≈ ¬a b ¬c = ordbij-refl b
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155 ... | tri> ¬a ¬b c = ⊥-elim ( be05 c iba iab )
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156
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157 _c<_ : ( A B : HOD ) → Set n
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158 A c< B = ¬ ( Injection (& A) (& B) )
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159
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160 Card : OD
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161 Card = record { def = λ x → (a : Ordinal) → a o< x → ¬ OrdBijection a x }
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162
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163 record Cardinal (a : Ordinal ) : Set (suc n) where
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164 field
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165 card : Ordinal
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166 ciso : OrdBijection a card
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167 cmax : (x : Ordinal) → card o< x → ¬ OrdBijection a x
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168
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169 Cardinal∈ : { s : HOD } → { t : Ordinal } → Ord t ∋ s → s c< Ord t
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170 Cardinal∈ = {!!}
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171
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172 Cardinal⊆ : { s t : HOD } → s ⊆ t → ( s c< t ) ∨ ( s =c= t )
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173 Cardinal⊆ = {!!}
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174
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175 Cantor1 : { u : HOD } → u c< Power u
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176 Cantor1 = {!!}
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177
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178 Cantor2 : { u : HOD } → ¬ ( u =c= Power u )
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179 Cantor2 = {!!}
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180
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182
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183
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