Theorem elwina 10700

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

Assertion

Ref	Expression
elwina	⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elwina

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ V)
2		fvex 6889	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2822	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 233	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1134	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → 𝐴 ∈ V)
6		neeq1 2994	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6876	. . . . 5 ⊢ (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8		eqeq12 2752	. . . . 5 ⊢ (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 688	. . . 4 ⊢ (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10		rexeq 3301	. . . . 5 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
11	10	raleqbi1dv 3317	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
12	6, 9, 11	3anbi123d 1438	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)))
13		df-wina 10698	. . 3 ⊢ Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3659	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)))
15	1, 5, 14	pm5.21nii 378	1 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))

Syntax hints:   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3459  ∅c0 4308   class class class wbr 5119  ‘cfv 6531   ≺ csdm 8958  cfccf 9951  Inacc_wcwina 10696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-wina 10698

This theorem is referenced by:  winaon  10702  inawina  10704  winacard  10706  winainf  10708  winalim2  10710  winafp  10711  gchina  10713