Theorem axpweq 4205

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4208 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 3620	. . . 4 |- ( ~P A e. _V -> ~P A e. ~P ~P A )
2		pweq 3609	. . . . . 6 |- ( x = ~P A -> ~P x = ~P ~P A )
3	2	eleq2d 2266	. . . . 5 |- ( x = ~P A -> ( ~P A e. ~P x <-> ~P A e. ~P ~P A ) )
4	3	spcegv 2852	. . . 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 2774	. . . 4 |- ( ~P A e. ~P x -> ~P A e. _V )
7	6	exlimiv 1612	. . 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 2766	. . . . 5 |- x e. _V
10	9	elpw2 4191	. . . 4 |- ( ~P A e. ~P x <-> ~P A C_ x )
11		pwss 3622	. . . . 5 |- ( ~P A C_ x <-> A. y ( y C_ A -> y e. x ) )
12		ssalel 3172	. . . . . . 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 1484	. . . . 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 1619	. 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 1362   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4152

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608

This theorem is referenced by: (None)