Theorem axpweq 4103

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4106 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 3529	. . . 4 |- ( ~P A e. _V -> ~P A e. ~P ~P A )
2		pweq 3518	. . . . . 6 |- ( x = ~P A -> ~P x = ~P ~P A )
3	2	eleq2d 2210	. . . . 5 |- ( x = ~P A -> ( ~P A e. ~P x <-> ~P A e. ~P ~P A ) )
4	3	spcegv 2777	. . . 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 2700	. . . 4 |- ( ~P A e. ~P x -> ~P A e. _V )
7	6	exlimiv 1578	. . 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 2692	. . . . 5 |- x e. _V
10	9	elpw2 4090	. . . 4 |- ( ~P A e. ~P x <-> ~P A C_ x )
11		pwss 3531	. . . . 5 |- ( ~P A C_ x <-> A. y ( y C_ A -> y e. x ) )
12		dfss2 3091	. . . . . . 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 1447	. . . . 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 1585	. 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 1330   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   C_ wss 3076   ~Pcpw 3515

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by: (None)