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Theorem pwex 3902
Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Hypothesis
Ref Expression
zfpowcl.1 A V
Assertion
Ref Expression
pwex 𝒫 A V

Proof of Theorem pwex
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfpowcl.1 . 2 A V
2 pweq 3333 . . 3 (z = A → 𝒫 z = 𝒫 A)
32eleq1d 2084 . 2 (z = A → (𝒫 z V ↔ 𝒫 A V))
4 df-pw 3332 . . 3 𝒫 z = {yyz}
5 axpow2 3899 . . . . . 6 xy(yzy x)
65bm1.3ii 3848 . . . . 5 xy(y xyz)
7 abeq2 2124 . . . . . 6 (x = {yyz} ↔ y(y xyz))
87exbii 1474 . . . . 5 (x x = {yyz} ↔ xy(y xyz))
96, 8mpbir 134 . . . 4 x x = {yyz}
109issetri 2538 . . 3 {yyz} V
114, 10eqeltri 2088 . 2 𝒫 z V
121, 3, 11vtocl 2581 1 𝒫 A V
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1224   = wceq 1226  wex 1358   wcel 1370  {cab 2004  Vcvv 2531  wss 2890  𝒫 cpw 3330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332
This theorem is referenced by:  pwexg  3903  p0ex  3909  pp0ex  3910  ord3ex  3911  abexssex  5671  npex  6321  axcnex  6549
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