MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  harcard Structured version   Visualization version   GIF version

Theorem harcard 8751
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 8413 . 2 (har‘𝐴) ∈ On
2 harndom 8416 . . . . . . 7 ¬ (har‘𝐴) ≼ 𝐴
3 simpll 789 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4 simpr 477 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5 elharval 8415 . . . . . . . . . . 11 (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
64, 5sylib 208 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
76simpld 475 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8 ontri1 5718 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
93, 7, 8syl2anc 692 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
10 simpllr 798 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≈ 𝑥)
11 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
12 ssdomg 7948 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥𝑦𝑥𝑦))
1311, 12ax-mp 5 . . . . . . . . . . 11 (𝑥𝑦𝑥𝑦)
146simprd 479 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝐴)
15 domtr 7956 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐴) → 𝑥𝐴)
1613, 14, 15syl2anr 495 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
17 endomtr 7961 . . . . . . . . . 10 (((har‘𝐴) ≈ 𝑥𝑥𝐴) → (har‘𝐴) ≼ 𝐴)
1810, 16, 17syl2anc 692 . . . . . . . . 9 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≼ 𝐴)
1918ex 450 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 → (har‘𝐴) ≼ 𝐴))
209, 19sylbird 250 . . . . . . 7 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦𝑥 → (har‘𝐴) ≼ 𝐴))
212, 20mt3i 141 . . . . . 6 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝑥)
2221ex 450 . . . . 5 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦𝑥))
2322ssrdv 3590 . . . 4 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
2423ex 450 . . 3 (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
2524rgen 2917 . 2 𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
26 iscard2 8749 . 2 ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
271, 25, 26mpbir2an 954 1 (card‘(har‘𝐴)) = (har‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3556   class class class wbr 4615  Oncon0 5684  cfv 5849  cen 7899  cdom 7900  harchar 8408  cardccrd 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-wrecs 7355  df-recs 7416  df-er 7690  df-en 7903  df-dom 7904  df-oi 8362  df-har 8410  df-card 8712
This theorem is referenced by:  cardprclem  8752  alephcard  8840  pwcfsdom  9352  hargch  9442
  Copyright terms: Public domain W3C validator