Theorem axpweq 4035

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4038 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

Hypothesis

Ref	Expression
axpweq.1	|- A e. _V

Assertion

Ref	Expression
axpweq	|- ( ~P A e. _V <-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )

Distinct variable group:   x, y, z, A

Proof of Theorem axpweq

Step	Hyp	Ref	Expression
1		pwidg 3471	. . . 4 |- ( ~P A e. _V -> ~P A e. ~P ~P A )
2		pweq 3460	. . . . . 6 |- ( x = ~P A -> ~P x = ~P ~P A )
3	2	eleq2d 2169	. . . . 5 |- ( x = ~P A -> ( ~P A e. ~P x <-> ~P A e. ~P ~P A ) )
4	3	spcegv 2729	. . . 4 |- ( ~P A e. _V -> ( ~P A e. ~P ~P A -> E. x ~P A e. ~P x ) )
5	1, 4	mpd 13	. . 3 |- ( ~P A e. _V -> E. x ~P A e. ~P x )
6		elex 2652	. . . 4 |- ( ~P A e. ~P x -> ~P A e. _V )
7	6	exlimiv 1545	. . 3 |- ( E. x ~P A e. ~P x -> ~P A e. _V )
8	5, 7	impbii 125	. 2 |- ( ~P A e. _V <-> E. x ~P A e. ~P x )
9		vex 2644	. . . . 5 |- x e. _V
10	9	elpw2 4022	. . . 4 |- ( ~P A e. ~P x <-> ~P A C_ x )
11		pwss 3473	. . . . 5 |- ( ~P A C_ x <-> A. y ( y C_ A -> y e. x ) )
12		dfss2 3036	. . . . . . 7 |- ( y C_ A <-> A. z ( z e. y -> z e. A ) )
13	12	imbi1i 237	. . . . . 6 |- ( ( y C_ A -> y e. x ) <-> ( A. z ( z e. y -> z e. A ) -> y e. x ) )
14	13	albii 1414	. . . . 5 |- ( A. y ( y C_ A -> y e. x ) <-> A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
15	11, 14	bitri 183	. . . 4 |- ( ~P A C_ x <-> A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
16	10, 15	bitri 183	. . 3 |- ( ~P A e. ~P x <-> A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
17	16	exbii 1552	. 2 |- ( E. x ~P A e. ~P x <-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
18	8, 17	bitri 183	1 |- ( ~P A e. _V <-> E. x A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1297   = wceq 1299   E.wex 1436   e. wcel 1448   _Vcvv 2641   C_ wss 3021   ~Pcpw 3457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459

This theorem is referenced by: (None)