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Theorem oncard 8641
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 6083 . . . . 5 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3 cardidm 8640 . . . . 5 (card‘(card‘𝑥)) = (card‘𝑥)
42, 3syl6eq 2654 . . . 4 (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
51, 4eqtr4d 2641 . . 3 (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
65exlimiv 1843 . 2 (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7 fvex 6093 . . . 4 (card‘𝐴) ∈ V
8 eleq1 2670 . . . 4 (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
97, 8mpbiri 246 . . 3 (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10 fveq2 6083 . . . . 5 (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
1110eqeq2d 2614 . . . 4 (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
1211spcegv 3261 . . 3 (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
139, 12mpcom 37 . 2 (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
146, 13impbii 197 1 (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wex 1694  wcel 1975  Vcvv 3167  cfv 5785  cardccrd 8616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-ord 5624  df-on 5625  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-er 7601  df-en 7814  df-card 8620
This theorem is referenced by: (None)
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