Theorem elwina 10772

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 3471	. 2 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ V)
2		fvex 6898	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2848	. . . 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 3017	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6885	. . . . 5 ⊢ (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8		eqeq12 2777	. . . . 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 3315	. . . . 5 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
11	10	raleqbi1dv 3329	. . . 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 10770	. . 3 ⊢ Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3633	. 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4278   class class class wbr 5102  ‘cfv 6538   ≺ csdm 8973  cfccf 10019  Inacc_wcwina 10768

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-wina 10770

This theorem is used by:  winaon  10774  inawina  10776  winacard  10778  winainf  10780  winalim2  10782  winafp  10783  gchina  10785