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Theorem harcard 10016
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 9597 . 2 (har‘𝐴) ∈ On
2 harndom 9600 . . . . . . 7 ¬ (har‘𝐴) ≼ 𝐴
3 simpll 767 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑥 ∈ On)
4 simpr 484 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ (har‘𝐴))
5 elharval 9599 . . . . . . . . . . 11 (𝑦 ∈ (har‘𝐴) ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
64, 5sylib 218 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑦 ∈ On ∧ 𝑦𝐴))
76simpld 494 . . . . . . . . 9 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦 ∈ On)
8 ontri1 6420 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
93, 7, 8syl2anc 584 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 ↔ ¬ 𝑦𝑥))
10 simpllr 776 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≈ 𝑥)
11 ssdomg 9039 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑥𝑦𝑥𝑦))
1211elv 3483 . . . . . . . . . . 11 (𝑥𝑦𝑥𝑦)
136simprd 495 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝐴)
14 domtr 9046 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐴) → 𝑥𝐴)
1512, 13, 14syl2anr 597 . . . . . . . . . 10 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
16 endomtr 9051 . . . . . . . . . 10 (((har‘𝐴) ≈ 𝑥𝑥𝐴) → (har‘𝐴) ≼ 𝐴)
1710, 15, 16syl2anc 584 . . . . . . . . 9 ((((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) ∧ 𝑥𝑦) → (har‘𝐴) ≼ 𝐴)
1817ex 412 . . . . . . . 8 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (𝑥𝑦 → (har‘𝐴) ≼ 𝐴))
199, 18sylbird 260 . . . . . . 7 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → (¬ 𝑦𝑥 → (har‘𝐴) ≼ 𝐴))
202, 19mt3i 149 . . . . . 6 (((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) ∧ 𝑦 ∈ (har‘𝐴)) → 𝑦𝑥)
2120ex 412 . . . . 5 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (𝑦 ∈ (har‘𝐴) → 𝑦𝑥))
2221ssrdv 4001 . . . 4 ((𝑥 ∈ On ∧ (har‘𝐴) ≈ 𝑥) → (har‘𝐴) ⊆ 𝑥)
2322ex 412 . . 3 (𝑥 ∈ On → ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥))
2423rgen 3061 . 2 𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)
25 iscard2 10014 . 2 ((card‘(har‘𝐴)) = (har‘𝐴) ↔ ((har‘𝐴) ∈ On ∧ ∀𝑥 ∈ On ((har‘𝐴) ≈ 𝑥 → (har‘𝐴) ⊆ 𝑥)))
261, 24, 25mpbir2an 711 1 (card‘(har‘𝐴)) = (har‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148  Oncon0 6386  cfv 6563  cen 8981  cdom 8982  harchar 9594  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-dom 8986  df-oi 9548  df-har 9595  df-card 9977
This theorem is referenced by:  cardprclem  10017  alephcard  10108  pwcfsdom  10621  hargch  10711
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