Theorem elwina 9508

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 3212	. 2 |- ( A e. InaccW -> A e. _V )
2		fvex 6201	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2689	. . . 4 |- ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4	2, 3	mpbii 223	. . 3 |- ( ( cf ` A ) = A -> A e. _V )
5	4	3ad2ant2 1083	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. _V )
6		neeq1 2856	. . . 4 |- ( z = A -> ( z =/= (/) <-> A =/= (/) ) )
7		fveq2 6191	. . . . 5 |- ( z = A -> ( cf ` z ) = ( cf ` A ) )
8		eqeq12 2635	. . . . 5 |- ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 703	. . . 4 |- ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10		rexeq 3139	. . . . 5 |- ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11	10	raleqbi1dv 3146	. . . 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 1399	. . 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 9506	. . 3 |- InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14	12, 13	elab2g 3353	. 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 368	1 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ` cfv 5888   ~< csdm 7954   cfccf 8763   InaccWcwina 9504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-wina 9506

This theorem is referenced by:  winaon  9510  inawina  9512  winacard  9514  winainf  9516  winalim2  9518  winafp  9519  gchina  9521