Theorem elwina 10726

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 3501	. 2 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ V)
2		fvex 6919	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2829	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 233	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1135	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → 𝐴 ∈ V)
6		neeq1 3003	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6906	. . . . 5 ⊢ (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8		eqeq12 2754	. . . . 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 3322	. . . . 5 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
11	10	raleqbi1dv 3338	. . . 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 10724	. . 3 ⊢ Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3680	. 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 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480  ∅c0 4333   class class class wbr 5143  ‘cfv 6561   ≺ csdm 8984  cfccf 9977  Inacc_wcwina 10722

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 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-wina 10724

This theorem is referenced by:  winaon  10728  inawina  10730  winacard  10732  winainf  10734  winalim2  10736  winafp  10737  gchina  10739