Theorem elwina 10160

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 3429	. 2 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ V)
2		fvex 6677	. . . 4 ⊢ (cf‘𝐴) ∈ V
3		eleq1 2840	. . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 236	. . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ V)
5	4	3ad2ant2 1132	. 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → 𝐴 ∈ V)
6		neeq1 3014	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7		fveq2 6664	. . . . 5 ⊢ (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8		eqeq12 2773	. . . . 5 ⊢ (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
9	7, 8	mpancom 687	. . . 4 ⊢ (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10		rexeq 3325	. . . . 5 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
11	10	raleqbi1dv 3322	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
12	6, 9, 11	3anbi123d 1434	. . 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 10158	. . 3 ⊢ Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑧 𝑥 ≺ 𝑦)}
14	12, 13	elab2g 3592	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)))
15	1, 5, 14	pm5.21nii 383	1 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))

Syntax hints:   ↔ wb 209   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  Vcvv 3410  ∅c0 4228   class class class wbr 5037  ‘cfv 6341   ≺ csdm 8540  cfccf 9413  Inacc_wcwina 10156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5181

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-iota 6300  df-fv 6349  df-wina 10158

This theorem is referenced by:  winaon  10162  inawina  10164  winacard  10166  winainf  10168  winalim2  10170  winafp  10171  gchina  10173