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Theorem harcard 10009
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 (cardβ€˜(harβ€˜π΄)) = (harβ€˜π΄)

Proof of Theorem harcard
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 harcl 9590 . 2 (harβ€˜π΄) ∈ On
2 harndom 9593 . . . . . . 7 Β¬ (harβ€˜π΄) β‰Ό 𝐴
3 simpll 765 . . . . . . . . 9 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ π‘₯ ∈ On)
4 simpr 483 . . . . . . . . . . 11 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 ∈ (harβ€˜π΄))
5 elharval 9592 . . . . . . . . . . 11 (𝑦 ∈ (harβ€˜π΄) ↔ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴))
64, 5sylib 217 . . . . . . . . . 10 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴))
76simpld 493 . . . . . . . . 9 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 ∈ On)
8 ontri1 6408 . . . . . . . . 9 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ π‘₯))
93, 7, 8syl2anc 582 . . . . . . . 8 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (π‘₯ βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ π‘₯))
10 simpllr 774 . . . . . . . . . 10 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ (harβ€˜π΄) β‰ˆ π‘₯)
11 ssdomg 9027 . . . . . . . . . . . 12 (𝑦 ∈ V β†’ (π‘₯ βŠ† 𝑦 β†’ π‘₯ β‰Ό 𝑦))
1211elv 3479 . . . . . . . . . . 11 (π‘₯ βŠ† 𝑦 β†’ π‘₯ β‰Ό 𝑦)
136simprd 494 . . . . . . . . . . 11 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 β‰Ό 𝐴)
14 domtr 9034 . . . . . . . . . . 11 ((π‘₯ β‰Ό 𝑦 ∧ 𝑦 β‰Ό 𝐴) β†’ π‘₯ β‰Ό 𝐴)
1512, 13, 14syl2anr 595 . . . . . . . . . 10 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ π‘₯ β‰Ό 𝐴)
16 endomtr 9039 . . . . . . . . . 10 (((harβ€˜π΄) β‰ˆ π‘₯ ∧ π‘₯ β‰Ό 𝐴) β†’ (harβ€˜π΄) β‰Ό 𝐴)
1710, 15, 16syl2anc 582 . . . . . . . . 9 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ (harβ€˜π΄) β‰Ό 𝐴)
1817ex 411 . . . . . . . 8 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (π‘₯ βŠ† 𝑦 β†’ (harβ€˜π΄) β‰Ό 𝐴))
199, 18sylbird 259 . . . . . . 7 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (Β¬ 𝑦 ∈ π‘₯ β†’ (harβ€˜π΄) β‰Ό 𝐴))
202, 19mt3i 149 . . . . . 6 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 ∈ π‘₯)
2120ex 411 . . . . 5 ((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) β†’ (𝑦 ∈ (harβ€˜π΄) β†’ 𝑦 ∈ π‘₯))
2221ssrdv 3988 . . . 4 ((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) β†’ (harβ€˜π΄) βŠ† π‘₯)
2322ex 411 . . 3 (π‘₯ ∈ On β†’ ((harβ€˜π΄) β‰ˆ π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
2423rgen 3060 . 2 βˆ€π‘₯ ∈ On ((harβ€˜π΄) β‰ˆ π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯)
25 iscard2 10007 . 2 ((cardβ€˜(harβ€˜π΄)) = (harβ€˜π΄) ↔ ((harβ€˜π΄) ∈ On ∧ βˆ€π‘₯ ∈ On ((harβ€˜π΄) β‰ˆ π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯)))
261, 24, 25mpbir2an 709 1 (cardβ€˜(harβ€˜π΄)) = (harβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   βŠ† wss 3949   class class class wbr 5152  Oncon0 6374  β€˜cfv 6553   β‰ˆ cen 8967   β‰Ό cdom 8968  harchar 9587  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-er 8731  df-en 8971  df-dom 8972  df-oi 9541  df-har 9588  df-card 9970
This theorem is referenced by:  cardprclem  10010  alephcard  10101  pwcfsdom  10614  hargch  10704
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