Theorem elwina 9129

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 3040	. 2 |- ( A e. InaccW -> A e. _V )
2		fvex 5889	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2537	. . . 4 |- ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4	2, 3	mpbii 216	. . 3 |- ( ( cf ` A ) = A -> A e. _V )
5	4	3ad2ant2 1052	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. _V )
6		neeq1 2705	. . . 4 |- ( z = A -> ( z =/= (/) <-> A =/= (/) ) )
7		fveq2 5879	. . . . 5 |- ( z = A -> ( cf ` z ) = ( cf ` A ) )
8		eqeq12 2484	. . . . 5 |- ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 682	. . . 4 |- ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10		rexeq 2974	. . . . 5 |- ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11	10	raleqbi1dv 2981	. . . 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 1365	. . 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 9127	. . 3 |- InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14	12, 13	elab2g 3175	. 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 360	1 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Syntax hints:   <-> wb 189   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   (/)c0 3722   class class class wbr 4395   ` cfv 5589   ~< csdm 7586   cfccf 8389   InaccWcwina 9125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-wina 9127

This theorem is referenced by:  winaon  9131  inawina  9133  winacard  9135  winainf  9137  winalim2  9139  winafp  9140  gchina  9142