Theorem oncard 10003

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝑥))
2		fveq2 6901	. . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3		cardidm 10002	. . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
4	2, 3	eqtrdi 2782	. . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
5	1, 4	eqtr4d 2769	. . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
6	5	exlimiv 1926	. 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7		fvex 6914	. . . 4 ⊢ (card‘𝐴) ∈ V
8		eleq1 2814	. . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
9	7, 8	mpbiri 257	. . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10		fveq2 6901	. . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
11	10	eqeq2d 2737	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
12	11	spcegv 3583	. . 3 ⊢ (𝐴 ∈ V → (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥)))
13	9, 12	mpcom 38	. 2 ⊢ (𝐴 = (card‘𝐴) → ∃𝑥 𝐴 = (card‘𝑥))
14	6, 13	impbii 208	1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))

Syntax hints:   ↔ wb 205   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3462  ‘cfv 6554  cardccrd 9978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  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 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  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-ord 6379  df-on 6380  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-er 8734  df-en 8975  df-card 9982

This theorem is referenced by:  iscard4  43200