Theorem winacard 9698

Description: A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)

Assertion

Ref	Expression
winacard	⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)

Proof of Theorem winacard

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elwina 9692	. 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
2		cardcf 9258	. . . 4 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)
3		fveq2 6344	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴))
4		id 22	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)
5	2, 3, 4	3eqtr3a 2810	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
6	5	3ad2ant2 1128	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → (card‘𝐴) = 𝐴)
7	1, 6	sylbi 207	1 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∀wral 3042  ∃wrex 3043  ∅c0 4050   class class class wbr 4796  ‘cfv 6041   ≺ csdm 8112  cardccrd 8943  cfccf 8945  Inacc_wcwina 9688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-er 7903  df-en 8114  df-card 8947  df-cf 8949  df-wina 9690

This theorem is referenced by:  winalim  9701  winalim2  9702  gchina  9705  inar1  9781