Theorem elina 9388

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 3185	. 2 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ V)
2		fvex 6113	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2676	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 222	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1076	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ V)
6		neeq1 2844	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6103	. . . . 5 ⊢ (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8		eqeq12 2623	. . . . 5 ⊢ (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 700	. . . 4 ⊢ (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10		breq2 4587	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝒫 𝑥 ≺ 𝑦 ↔ 𝒫 𝑥 ≺ 𝐴))
11	10	raleqbi1dv 3123	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
12	6, 9, 11	3anbi123d 1391	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
13		df-ina 9386	. . 3 ⊢ Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥 ∈ 𝑦 𝒫 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3322	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)))
15	1, 5, 14	pm5.21nii 367	1 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))

Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804   ≺ csdm 7840  cfccf 8646  Inacccina 9384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ina 9386

This theorem is referenced by:  inawina  9391  omina  9392  gchina  9400  inar1  9476  inatsk  9479  tskcard  9482  tskuni  9484  gruina  9519  grur1  9521