Theorem oncard 10029

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 6920	. . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3		cardidm 10028	. . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
4	2, 3	eqtrdi 2796	. . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
5	1, 4	eqtr4d 2783	. . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
6	5	exlimiv 1929	. 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7		fvex 6933	. . . 4 ⊢ (card‘𝐴) ∈ V
8		eleq1 2832	. . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
9	7, 8	mpbiri 258	. . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
11	10	eqeq2d 2751	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
12	11	spcegv 3610	. . 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 209	1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  ‘cfv 6573  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-er 8763  df-en 9004  df-card 10008

This theorem is referenced by:  iscard4  43495