Theorem oncard 9576

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 6717	. . . . 5 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘(card‘𝑥)))
3		cardidm 9575	. . . . 5 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
4	2, 3	eqtrdi 2794	. . . 4 ⊢ (𝐴 = (card‘𝑥) → (card‘𝐴) = (card‘𝑥))
5	1, 4	eqtr4d 2780	. . 3 ⊢ (𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
6	5	exlimiv 1938	. 2 ⊢ (∃𝑥 𝐴 = (card‘𝑥) → 𝐴 = (card‘𝐴))
7		fvex 6730	. . . 4 ⊢ (card‘𝐴) ∈ V
8		eleq1 2825	. . . 4 ⊢ (𝐴 = (card‘𝐴) → (𝐴 ∈ V ↔ (card‘𝐴) ∈ V))
9	7, 8	mpbiri 261	. . 3 ⊢ (𝐴 = (card‘𝐴) → 𝐴 ∈ V)
10		fveq2 6717	. . . . 5 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴))
11	10	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴)))
12	11	spcegv 3512	. . 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 212	1 ⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))

Syntax hints:   ↔ wb 209   = wceq 1543  ∃wex 1787   ∈ wcel 2110  Vcvv 3408  ‘cfv 6380  cardccrd 9551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-er 8391  df-en 8627  df-card 9555

This theorem is referenced by:  iscard4  40825