Theorem elina 9494

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 3207	. 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V)
2		fvex 6188	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2687	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 223	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1081	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V)
6		neeq1 2853	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6178	. . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8		eqeq12 2633	. . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 702	. . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10		breq2 4648	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴))
11	10	raleqbi1dv 3141	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
12	6, 9, 11	3anbi123d 1397	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
13		df-ina 9492	. . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3347	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
15	1, 5, 14	pm5.21nii 368	1 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))

Syntax hints:   ↔ wb 196   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  Vcvv 3195  ∅c0 3907  𝒫 cpw 4149   class class class wbr 4644  ‘cfv 5876   ≺ csdm 7939  cfccf 8748  Inacccina 9490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ina 9492

This theorem is referenced by:  inawina  9497  omina  9498  gchina  9506  inar1  9582  inatsk  9585  tskcard  9588  tskuni  9590  gruina  9625  grur1  9627