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Theorem harcard 8687
 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 8349 . 2 (har‘𝐴) ∈ On
2 harndom 8352 . . . . . . 7 ¬ (har‘𝐴) ≼ 𝐴
3 simpll 786 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4 simpr 476 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5 elharval 8351 . . . . . . . . . . 11 (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
64, 5sylib 207 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
76simpld 474 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8 ontri1 5674 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
93, 7, 8syl2anc 691 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
10 simpllr 795 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≈ 𝑥)
11 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
12 ssdomg 7887 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥𝑦𝑥𝑦))
1311, 12ax-mp 5 . . . . . . . . . . 11 (𝑥𝑦𝑥𝑦)
146simprd 478 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝐴)
15 domtr 7895 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐴) → 𝑥𝐴)
1613, 14, 15syl2anr 494 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
17 endomtr 7900 . . . . . . . . . 10 (((har‘𝐴) ≈ 𝑥𝑥𝐴) → (har‘𝐴) ≼ 𝐴)
1810, 16, 17syl2anc 691 . . . . . . . . 9 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≼ 𝐴)
1918ex 449 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 → (har‘𝐴) ≼ 𝐴))
209, 19sylbird 249 . . . . . . 7 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦𝑥 → (har‘𝐴) ≼ 𝐴))
212, 20mt3i 140 . . . . . 6 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝑥)
2221ex 449 . . . . 5 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦𝑥))
2322ssrdv 3574 . . . 4 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
2423ex 449 . . 3 (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
2524rgen 2906 . 2 𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
26 iscard2 8685 . 2 ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
271, 25, 26mpbir2an 957 1 (card‘(har‘𝐴)) = (har‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  Oncon0 5640  ‘cfv 5804   ≈ cen 7838   ≼ cdom 7839  harchar 8344  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-oi 8298  df-har 8346  df-card 8648 This theorem is referenced by:  cardprclem  8688  alephcard  8776  pwcfsdom  9284  hargch  9374
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