Theorem winacard 10106

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 10100	. 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
2		cardcf 9666	. . . 4 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)
3		fveq2 6663	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (card‘(cf‘𝐴)) = (card‘𝐴))
4		id 22	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)
5	2, 3, 4	3eqtr3a 2878	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → (card‘𝐴) = 𝐴)
6	5	3ad2ant2 1129	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → (card‘𝐴) = 𝐴)
7	1, 6	sylbi 219	1 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  ∅c0 4289   class class class wbr 5057  ‘cfv 6348   ≺ csdm 8500  cardccrd 9356  cfccf 9358  Inacc_wcwina 10096

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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8281  df-en 8502  df-card 9360  df-cf 9362  df-wina 10098

This theorem is referenced by:  winalim  10109  winalim2  10110  gchina  10113  inar1  10189