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Theorem harcard 9972
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 9553 . 2 (harβ€˜π΄) ∈ On
2 harndom 9556 . . . . . . 7 Β¬ (harβ€˜π΄) β‰Ό 𝐴
3 simpll 764 . . . . . . . . 9 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ π‘₯ ∈ On)
4 simpr 484 . . . . . . . . . . 11 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 ∈ (harβ€˜π΄))
5 elharval 9555 . . . . . . . . . . 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 494 . . . . . . . . 9 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 ∈ On)
8 ontri1 6391 . . . . . . . . 9 ((π‘₯ ∈ On ∧ 𝑦 ∈ On) β†’ (π‘₯ βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ π‘₯))
93, 7, 8syl2anc 583 . . . . . . . 8 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (π‘₯ βŠ† 𝑦 ↔ Β¬ 𝑦 ∈ π‘₯))
10 simpllr 773 . . . . . . . . . 10 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ (harβ€˜π΄) β‰ˆ π‘₯)
11 ssdomg 8995 . . . . . . . . . . . 12 (𝑦 ∈ V β†’ (π‘₯ βŠ† 𝑦 β†’ π‘₯ β‰Ό 𝑦))
1211elv 3474 . . . . . . . . . . 11 (π‘₯ βŠ† 𝑦 β†’ π‘₯ β‰Ό 𝑦)
136simprd 495 . . . . . . . . . . 11 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ 𝑦 β‰Ό 𝐴)
14 domtr 9002 . . . . . . . . . . 11 ((π‘₯ β‰Ό 𝑦 ∧ 𝑦 β‰Ό 𝐴) β†’ π‘₯ β‰Ό 𝐴)
1512, 13, 14syl2anr 596 . . . . . . . . . 10 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ π‘₯ β‰Ό 𝐴)
16 endomtr 9007 . . . . . . . . . 10 (((harβ€˜π΄) β‰ˆ π‘₯ ∧ π‘₯ β‰Ό 𝐴) β†’ (harβ€˜π΄) β‰Ό 𝐴)
1710, 15, 16syl2anc 583 . . . . . . . . 9 ((((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) ∧ π‘₯ βŠ† 𝑦) β†’ (harβ€˜π΄) β‰Ό 𝐴)
1817ex 412 . . . . . . . 8 (((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) ∧ 𝑦 ∈ (harβ€˜π΄)) β†’ (π‘₯ βŠ† 𝑦 β†’ (harβ€˜π΄) β‰Ό 𝐴))
199, 18sylbird 260 . . . . . . 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 412 . . . . 5 ((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) β†’ (𝑦 ∈ (harβ€˜π΄) β†’ 𝑦 ∈ π‘₯))
2221ssrdv 3983 . . . 4 ((π‘₯ ∈ On ∧ (harβ€˜π΄) β‰ˆ π‘₯) β†’ (harβ€˜π΄) βŠ† π‘₯)
2322ex 412 . . 3 (π‘₯ ∈ On β†’ ((harβ€˜π΄) β‰ˆ π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯))
2423rgen 3057 . 2 βˆ€π‘₯ ∈ On ((harβ€˜π΄) β‰ˆ π‘₯ β†’ (harβ€˜π΄) βŠ† π‘₯)
25 iscard2 9970 . 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 708 1 (cardβ€˜(harβ€˜π΄)) = (harβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141  Oncon0 6357  β€˜cfv 6536   β‰ˆ cen 8935   β‰Ό cdom 8936  harchar 9550  cardccrd 9929
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-er 8702  df-en 8939  df-dom 8940  df-oi 9504  df-har 9551  df-card 9933
This theorem is referenced by:  cardprclem  9973  alephcard  10064  pwcfsdom  10577  hargch  10667
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