Theorem elwina 9111

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 3090	. 2 |- ( A e. InaccW -> A e. _V )
2		fvex 5887	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2494	. . . 4 |- ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4	2, 3	mpbii 214	. . 3 |- ( ( cf ` A ) = A -> A e. _V )
5	4	3ad2ant2 1027	. 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 5877	. . . . 5 |- ( z = A -> ( cf ` z ) = ( cf ` A ) )
8		eqeq12 2441	. . . . 5 |- ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 673	. . . 4 |- ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10		rexeq 3026	. . . . 5 |- ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11	10	raleqbi1dv 3033	. . . 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 1335	. . 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 9109	. . 3 |- InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14	12, 13	elab2g 3220	. 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 354	1 |- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Syntax hints:   <-> wb 187   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081   (/)c0 3761   class class class wbr 4420   ` cfv 5597   ~< csdm 7572   cfccf 8372   InaccWcwina 9107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-nul 4551

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-iota 5561  df-fv 5605  df-wina 9109

This theorem is referenced by:  winaon  9113  inawina  9115  winacard  9117  winainf  9119  winalim2  9121  winafp  9122  gchina  9124