Theorem axpweq 4150

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4153 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 3573	. . . 4 |- ( ~P A e. _V -> ~P A e. ~P ~P A )
2		pweq 3562	. . . . . 6 |- ( x = ~P A -> ~P x = ~P ~P A )
3	2	eleq2d 2236	. . . . 5 |- ( x = ~P A -> ( ~P A e. ~P x <-> ~P A e. ~P ~P A ) )
4	3	spcegv 2814	. . . 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 2737	. . . 4 |- ( ~P A e. ~P x -> ~P A e. _V )
7	6	exlimiv 1586	. . 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 2729	. . . . 5 |- x e. _V
10	9	elpw2 4136	. . . 4 |- ( ~P A e. ~P x <-> ~P A C_ x )
11		pwss 3575	. . . . 5 |- ( ~P A C_ x <-> A. y ( y C_ A -> y e. x ) )
12		dfss2 3131	. . . . . . 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 1458	. . . . 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 1593	. 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 1341   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by: (None)