Theorem elina 9944

Description: Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)

Assertion

Ref	Expression
elina	⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem elina

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V)
2		fvex 6543	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2868	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 234	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1125	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V)
6		neeq1 3044	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6530	. . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8		eqeq12 2806	. . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 684	. . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10		breq2 4960	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴))
11	10	raleqbi1dv 3360	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
12	6, 9, 11	3anbi123d 1426	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
13		df-ina 9942	. . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3602	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
15	1, 5, 14	pm5.21nii 380	1 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))

Syntax hints:   ↔ wb 207   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079   ≠ wne 2982  ∀wral 3103  Vcvv 3432  ∅c0 4206  𝒫 cpw 4447   class class class wbr 4956  ‘cfv 6217   ≺ csdm 8346  cfccf 9201  Inacccina 9940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767  ax-nul 5095

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225  df-ina 9942

This theorem is referenced by:  inawina  9947  omina  9948  gchina  9956  inar1  10032  inatsk  10035  tskcard  10038  tskuni  10040  gruina  10075  grur1  10077