Theorem elina 9094

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 3067	. 2 |- ( A e. Inacc -> A e. _V )
2		fvex 5858	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2474	. . . 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 1019	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) -> A e. _V )
6		neeq1 2684	. . . 4 |- ( y = A -> ( y =/= (/) <-> A =/= (/) ) )
7		fveq2 5848	. . . . 5 |- ( y = A -> ( cf ` y ) = ( cf ` A ) )
8		eqeq12 2421	. . . . 5 |- ( ( ( cf ` y ) = ( cf ` A ) /\ y = A ) -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 667	. . . 4 |- ( y = A -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
10		breq2 4398	. . . . 5 |- ( y = A -> ( ~P x ~< y <-> ~P x ~< A ) )
11	10	raleqbi1dv 3011	. . . 4 |- ( y = A -> ( A. x e. y ~P x ~< y <-> A. x e. A ~P x ~< A ) )
12	6, 9, 11	3anbi123d 1301	. . 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 9092	. . 3 |- Inacc = { y | ( y =/= (/) /\ ( cf ` y ) = y /\ A. x e. y ~P x ~< y ) }
14	12, 13	elab2g 3197	. 2 |- ( A e. _V -> ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) ) )
15	1, 5, 14	pm5.21nii 351	1 |- ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) )

Syntax hints:   <-> wb 184   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2753   _Vcvv 3058   (/)c0 3737   ~Pcpw 3954   class class class wbr 4394   ` cfv 5568   ~< csdm 7552   cfccf 8349   Inacccina 9090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ina 9092

This theorem is referenced by:  inawina  9097  omina  9098  gchina  9106  inar1  9182  inatsk  9185  tskcard  9188  tskuni  9190  gruina  9225  grur1  9227