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Theorem harcard 9090
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 8708 . 2 (har‘𝐴) ∈ On
2 harndom 8711 . . . . . . 7 ¬ (har‘𝐴) ≼ 𝐴
3 simpll 784 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4 simpr 478 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5 elharval 8710 . . . . . . . . . . 11 (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
64, 5sylib 210 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
76simpld 489 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8 ontri1 5975 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
93, 7, 8syl2anc 580 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
10 simpllr 794 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≈ 𝑥)
11 vex 3388 . . . . . . . . . . . 12 𝑦 ∈ V
12 ssdomg 8241 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥𝑦𝑥𝑦))
1311, 12ax-mp 5 . . . . . . . . . . 11 (𝑥𝑦𝑥𝑦)
146simprd 490 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝐴)
15 domtr 8248 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐴) → 𝑥𝐴)
1613, 14, 15syl2anr 591 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
17 endomtr 8253 . . . . . . . . . 10 (((har‘𝐴) ≈ 𝑥𝑥𝐴) → (har‘𝐴) ≼ 𝐴)
1810, 16, 17syl2anc 580 . . . . . . . . 9 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≼ 𝐴)
1918ex 402 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 → (har‘𝐴) ≼ 𝐴))
209, 19sylbird 252 . . . . . . 7 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦𝑥 → (har‘𝐴) ≼ 𝐴))
212, 20mt3i 144 . . . . . 6 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝑥)
2221ex 402 . . . . 5 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦𝑥))
2322ssrdv 3804 . . . 4 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
2423ex 402 . . 3 (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
2524rgen 3103 . 2 𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
26 iscard2 9088 . 2 ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
271, 25, 26mpbir2an 703 1 (card‘(har‘𝐴)) = (har‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  wss 3769   class class class wbr 4843  Oncon0 5941  cfv 6101  cen 8192  cdom 8193  harchar 8703  cardccrd 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-wrecs 7645  df-recs 7707  df-er 7982  df-en 8196  df-dom 8197  df-oi 8657  df-har 8705  df-card 9051
This theorem is referenced by:  cardprclem  9091  alephcard  9179  pwcfsdom  9693  hargch  9783
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