Theorem elina 9112

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

Assertion

Ref	Expression
elina	|- ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) )

Distinct variable group:   x, A

Proof of Theorem elina

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3054	. 2 |- ( A e. Inacc -> A e. _V )
2		fvex 5875	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2517	. . . 4 |- ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4	2, 3	mpbii 215	. . 3 |- ( ( cf ` A ) = A -> A e. _V )
5	4	3ad2ant2 1030	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) -> A e. _V )
6		neeq1 2686	. . . 4 |- ( y = A -> ( y =/= (/) <-> A =/= (/) ) )
7		fveq2 5865	. . . . 5 |- ( y = A -> ( cf ` y ) = ( cf ` A ) )
8		eqeq12 2464	. . . . 5 |- ( ( ( cf ` y ) = ( cf ` A ) /\ y = A ) -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 675	. . . 4 |- ( y = A -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
10		breq2 4406	. . . . 5 |- ( y = A -> ( ~P x ~< y <-> ~P x ~< A ) )
11	10	raleqbi1dv 2995	. . . 4 |- ( y = A -> ( A. x e. y ~P x ~< y <-> A. x e. A ~P x ~< A ) )
12	6, 9, 11	3anbi123d 1339	. . 3 |- ( y = A -> ( ( y =/= (/) /\ ( cf ` y ) = y /\ A. x e. y ~P x ~< y ) <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) ) )
13		df-ina 9110	. . 3 |- Inacc = { y | ( y =/= (/) /\ ( cf ` y ) = y /\ A. x e. y ~P x ~< y ) }
14	12, 13	elab2g 3187	. 2 |- ( A e. _V -> ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) ) )
15	1, 5, 14	pm5.21nii 355	1 |- ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) )

Syntax hints:   <-> wb 188   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   _Vcvv 3045   (/)c0 3731   ~Pcpw 3951   class class class wbr 4402   ` cfv 5582   ~< csdm 7568   cfccf 8371   Inacccina 9108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-ina 9110

This theorem is referenced by:  inawina  9115  omina  9116  gchina  9124  inar1  9200  inatsk  9203  tskcard  9206  tskuni  9208  gruina  9243  grur1  9245