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Theorem oncard 10001
Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
oncard (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem oncard
StepHypRef Expression
1 id 22 . . . 4 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
2 fveq2 6905 . . . . 5 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3 cardidm 10000 . . . . 5 (card‘(card‘𝑥)) = (card‘𝑥)
42, 3eqtrdi 2792 . . . 4 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
51, 4eqtr4d 2779 . . 3 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
65exlimiv 1929 . 2 (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7 fvex 6918 . . . 4 (card‘𝐴) ∈ V
8 eleq1 2828 . . . 4 (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
97, 8mpbiri 258 . . 3 (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10 fveq2 6905 . . . . 5 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1110eqeq2d 2747 . . . 4 (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
1211spcegv 3596 . . 3 (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
139, 12mpcom 38 . 2 (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
146, 13impbii 209 1 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  cfv 6560  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-er 8746  df-en 8987  df-card 9980
This theorem is referenced by:  iscard4  43551
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