Theorem elwina 10698

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 3474	. 2 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ V)
2		fvex 6895	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2850	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 236	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1152	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → 𝐴 ∈ V)
6		neeq1 3019	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6882	. . . . 5 ⊢ (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8		eqeq12 2779	. . . . 5 ⊢ (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 701	. . . 4 ⊢ (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10		rexeq 3317	. . . . 5 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
11	10	raleqbi1dv 3331	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
12	6, 9, 11	3anbi123d 1464	. . 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 10696	. . 3 ⊢ Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3637	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)))
15	1, 5, 14	pm5.21nii 381	1 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3453  ∅c0 4282   class class class wbr 5107  ‘cfv 6537   ≺ csdm 8954  cfccf 9945  Inacc_wcwina 10694

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-wina 10696

This theorem is used by:  winaon  10700  inawina  10702  winacard  10704  winainf  10706  winalim2  10708  winafp  10709  gchina  10711