Theorem elina 8846

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 2976	. 2 |- ( A e. Inacc -> A e. _V )
2		fvex 5696	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2498	. . . 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 1010	. 2 |- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) -> A e. _V )
6		neeq1 2611	. . . 4 |- ( y = A -> ( y =/= (/) <-> A =/= (/) ) )
7		fveq2 5686	. . . . 5 |- ( y = A -> ( cf ` y ) = ( cf ` A ) )
8		eqeq12 2450	. . . . 5 |- ( ( ( cf ` y ) = ( cf ` A ) /\ y = A ) -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
9	7, 8	mpancom 669	. . . 4 |- ( y = A -> ( ( cf ` y ) = y <-> ( cf ` A ) = A ) )
10		breq2 4291	. . . . 5 |- ( y = A -> ( ~P x ~< y <-> ~P x ~< A ) )
11	10	raleqbi1dv 2920	. . . 4 |- ( y = A -> ( A. x e. y ~P x ~< y <-> A. x e. A ~P x ~< A ) )
12	6, 9, 11	3anbi123d 1289	. . 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 8844	. . 3 |- Inacc = { y | ( y =/= (/) /\ ( cf ` y ) = y /\ A. x e. y ~P x ~< y ) }
14	12, 13	elab2g 3103	. 2 |- ( A e. _V -> ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) ) )
15	1, 5, 14	pm5.21nii 353	1 |- ( A e. Inacc <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A ~P x ~< A ) )

Syntax hints:   <-> wb 184   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   _Vcvv 2967   (/)c0 3632   ~Pcpw 3855   class class class wbr 4287   ` cfv 5413   ~< csdm 7301   cfccf 8099   Inacccina 8842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ina 8844

This theorem is referenced by:  inawina  8849  omina  8850  gchina  8858  inar1  8934  inatsk  8937  tskcard  8940  tskuni  8942  gruina  8977  grur1  8979