Theorem elwina 9053

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

Assertion

Ref	Expression
elwina	|- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. 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 3115	. 2 |- ( A e. InaccW -> A e. _V )
2		fvex 5858	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2526	. . . 4 |- ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4	2, 3	mpbii 211	. . 3 |- ( ( cf ` A ) = A -> A e. _V )
5	4	3ad2ant2 1016	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. _V )
6		neeq1 2735	. . . 4 |- ( z = A -> ( z =/= (/) <-> A =/= (/) ) )
7		fveq2 5848	. . . . 5 |- ( z = A -> ( cf ` z ) = ( cf ` A ) )
8		eqeq12 2473	. . . . 5 |- ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 667	. . . 4 |- ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10		rexeq 3052	. . . . 5 |- ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11	10	raleqbi1dv 3059	. . . 4 |- ( z = A -> ( A. x e. z E. y e. z x ~< y <-> A. x e. A E. y e. A x ~< y ) )
12	6, 9, 11	3anbi123d 1297	. . 3 |- ( z = A -> ( ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) ) )
13		df-wina 9051	. . 3 |- InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14	12, 13	elab2g 3245	. 2 |- ( A e. _V -> ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) ) )
15	1, 5, 14	pm5.21nii 351	1 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Syntax hints:   <-> wb 184   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   ` cfv 5570   ~< csdm 7508   cfccf 8309   InaccWcwina 9049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-wina 9051

This theorem is referenced by:  winaon  9055  inawina  9057  winacard  9059  winainf  9061  winalim2  9063  winafp  9064  gchina  9066