Theorem elina 8968

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 3087	. 2 |- ( A e. Inacc -> A e. _V )
2		fvex 5812	. . . 4 |- ( cf ` A ) e. _V
3		eleq1 2526	. . . 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 2733	. . . 4 |- ( y = A -> ( y =/= (/) <-> A =/= (/) ) )
7		fveq2 5802	. . . . 5 |- ( y = A -> ( cf ` y ) = ( cf ` A ) )
8		eqeq12 2473	. . . . 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 4407	. . . . 5 |- ( y = A -> ( ~P x ~< y <-> ~P x ~< A ) )
11	10	raleqbi1dv 3031	. . . 4 |- ( y = A -> ( A. x e. y ~P x ~< y <-> A. x e. A ~P x ~< A ) )
12	6, 9, 11	3anbi123d 1290	. . 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 8966	. . 3 |- Inacc = { y | ( y =/= (/) /\ ( cf ` y ) = y /\ A. x e. y ~P x ~< y ) }
14	12, 13	elab2g 3215	. 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 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   _Vcvv 3078   (/)c0 3748   ~Pcpw 3971   class class class wbr 4403   ` cfv 5529   ~< csdm 7422   cfccf 8221   Inacccina 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ina 8966

This theorem is referenced by:  inawina  8971  omina  8972  gchina  8980  inar1  9056  inatsk  9059  tskcard  9062  tskuni  9064  gruina  9099  grur1  9101