Theorem axpweq 4223

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4226 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 3635	. . . 4 |- ( ~P A e. _V -> ~P A e. ~P ~P A )
2		pweq 3624	. . . . . 6 |- ( x = ~P A -> ~P x = ~P ~P A )
3	2	eleq2d 2276	. . . . 5 |- ( x = ~P A -> ( ~P A e. ~P x <-> ~P A e. ~P ~P A ) )
4	3	spcegv 2865	. . . 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 2785	. . . 4 |- ( ~P A e. ~P x -> ~P A e. _V )
7	6	exlimiv 1622	. . 3 |- ( E. x ~P A e. ~P x -> ~P A e. _V )
8	5, 7	impbii 126	. 2 |- ( ~P A e. _V <-> E. x ~P A e. ~P x )
9		vex 2776	. . . . 5 |- x e. _V
10	9	elpw2 4209	. . . 4 |- ( ~P A e. ~P x <-> ~P A C_ x )
11		pwss 3637	. . . . 5 |- ( ~P A C_ x <-> A. y ( y C_ A -> y e. x ) )
12		ssalel 3185	. . . . . . 7 |- ( y C_ A <-> A. z ( z e. y -> z e. A ) )
13	12	imbi1i 238	. . . . . 6 |- ( ( y C_ A -> y e. x ) <-> ( A. z ( z e. y -> z e. A ) -> y e. x ) )
14	13	albii 1494	. . . . 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 184	. . . 4 |- ( ~P A C_ x <-> A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
16	10, 15	bitri 184	. . 3 |- ( ~P A e. ~P x <-> A. y ( A. z ( z e. y -> z e. A ) -> y e. x ) )
17	16	exbii 1629	. 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 184	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 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   C_ wss 3170   ~Pcpw 3621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623

This theorem is referenced by: (None)