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Theorem oncard 9954
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 6884 . . . . 5 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3 cardidm 9953 . . . . 5 (card‘(card‘𝑥)) = (card‘𝑥)
42, 3eqtrdi 2782 . . . 4 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
51, 4eqtr4d 2769 . . 3 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
65exlimiv 1925 . 2 (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7 fvex 6897 . . . 4 (card‘𝐴) ∈ V
8 eleq1 2815 . . . 4 (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
97, 8mpbiri 258 . . 3 (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10 fveq2 6884 . . . . 5 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1110eqeq2d 2737 . . . 4 (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
1211spcegv 3581 . . 3 (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
139, 12mpcom 38 . 2 (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
146, 13impbii 208 1 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cfv 6536  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-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-rab 3427  df-v 3470  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-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-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-ord 6360  df-on 6361  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-er 8702  df-en 8939  df-card 9933
This theorem is referenced by:  iscard4  42841
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