Theorem elwina 8954

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 3077	. 2 |- ( A e. InaccW -> A e. _V )
2		fvex 5799	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2523	. . . 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 1010	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. _V )
6		neeq1 2729	. . . 4 |- ( z = A -> ( z =/= (/) <-> A =/= (/) ) )
7		fveq2 5789	. . . . 5 |- ( z = A -> ( cf ` z ) = ( cf ` A ) )
8		eqeq12 2470	. . . . 5 |- ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 669	. . . 4 |- ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10		rexeq 3014	. . . . 5 |- ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11	10	raleqbi1dv 3021	. . . 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 1290	. . 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 8952	. . 3 |- InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14	12, 13	elab2g 3205	. 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 353	1 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Syntax hints:   <-> wb 184   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   E.wrex 2796   _Vcvv 3068   (/)c0 3735   class class class wbr 4390   ` cfv 5516   ~< csdm 7409   cfccf 8208   InaccWcwina 8950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4519

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-wina 8952

This theorem is referenced by:  winaon  8956  inawina  8958  winacard  8960  winainf  8962  winalim2  8964  winafp  8965  gchina  8967