Theorem axpweq 3480

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3481 is not used by the proof.

Hypothesis

Ref	Expression
axpweq.1	|- A e. _V

Assertion

Ref	Expression
axpweq	|- (~PA e. _V <-> E.xA.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		ssid 2634	. . . . 5 |- ~PA C_ ~PA
2		elpwg 3038	. . . . 5 |- (~PA e. _V -> (~PA e. ~P~PA <-> ~PA C_ ~PA))
3	1, 2	mpbiri 211	. . . 4 |- (~PA e. _V -> ~PA e. ~P~PA)
4		pweq 3036	. . . . . 6 |- (x = ~PA -> ~Px = ~P~PA)
5	4	eleq2d 1964	. . . . 5 |- (x = ~PA -> (~PA e. ~Px <-> ~PA e. ~P~PA))
6	5	cla4egv 2365	. . . 4 |- (~PA e. _V -> (~PA e. ~P~PA -> E.x~PA e. ~Px))
7	3, 6	mpd 29	. . 3 |- (~PA e. _V -> E.x~PA e. ~Px)
8		elisset 2299	. . . 4 |- (~PA e. ~Px -> ~PA e. _V)
9	8	19.23aiv 1674	. . 3 |- (E.x~PA e. ~Px -> ~PA e. _V)
10	7, 9	impbii 174	. 2 |- (~PA e. _V <-> E.x~PA e. ~Px)
11		visset 2295	. . . . 5 |- x e. _V
12	11	elpw2 3464	. . . 4 |- (~PA e. ~Px <-> ~PA C_ x)
13		pwss 3043	. . . 4 |- (~PA C_ x <-> A.y(y C_ A -> y e. x))
14		dfss2 2610	. . . . . 6 |- (y C_ A <-> A.z(z e. y -> z e. A))
15	14	imbi1i 203	. . . . 5 |- ((y C_ A -> y e. x) <-> (A.z(z e. y -> z e. A) -> y e. x))
16	15	albii 1346	. . . 4 |- (A.y(y C_ A -> y e. x) <-> A.y(A.z(z e. y -> z e. A) -> y e. x))
17	12, 13, 16	3bitri 194	. . 3 |- (~PA e. ~Px <-> A.y(A.z(z e. y -> z e. A) -> y e. x))
18	17	exbii 1398	. 2 |- (E.x~PA e. ~Px <-> E.xA.y(A.z(z e. y -> z e. A) -> y e. x))
19	10, 18	bitri 190	1 |- (~PA e. _V <-> E.xA.y(A.z(z e. y -> z e. A) -> y e. x))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035

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