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Theorem harcard 9404
 Description: The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
harcard (card‘(har‘𝐴)) = (har‘𝐴)

Proof of Theorem harcard
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 harcl 9020 . 2 (har‘𝐴) ∈ On
2 harndom 9023 . . . . . . 7 ¬ (har‘𝐴) ≼ 𝐴
3 simpll 766 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4 simpr 488 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5 elharval 9022 . . . . . . . . . . 11 (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
64, 5sylib 221 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
76simpld 498 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8 ontri1 6212 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
93, 7, 8syl2anc 587 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
10 simpllr 775 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≈ 𝑥)
11 ssdomg 8551 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥𝑦𝑥𝑦))
1211elv 3485 . . . . . . . . . . 11 (𝑥𝑦𝑥𝑦)
136simprd 499 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝐴)
14 domtr 8558 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐴) → 𝑥𝐴)
1512, 13, 14syl2anr 599 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
16 endomtr 8563 . . . . . . . . . 10 (((har‘𝐴) ≈ 𝑥𝑥𝐴) → (har‘𝐴) ≼ 𝐴)
1710, 15, 16syl2anc 587 . . . . . . . . 9 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≼ 𝐴)
1817ex 416 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 → (har‘𝐴) ≼ 𝐴))
199, 18sylbird 263 . . . . . . 7 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦𝑥 → (har‘𝐴) ≼ 𝐴))
202, 19mt3i 151 . . . . . 6 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝑥)
2120ex 416 . . . . 5 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦𝑥))
2221ssrdv 3959 . . . 4 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
2322ex 416 . . 3 (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
2423rgen 3143 . 2 𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
25 iscard2 9402 . 2 ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
261, 24, 25mpbir2an 710 1 (card‘(har‘𝐴)) = (har‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ⊆ wss 3919   class class class wbr 5052  Oncon0 6178  ‘cfv 6343   ≈ cen 8502   ≼ cdom 8503  harchar 9017  cardccrd 9361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-wrecs 7943  df-recs 8004  df-er 8285  df-en 8506  df-dom 8507  df-oi 8971  df-har 9018  df-card 9365 This theorem is referenced by:  cardprclem  9405  alephcard  9494  pwcfsdom  10003  hargch  10093
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