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Theorem pwex 3923
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 3354 . . 3 (z = A → 𝒫 z = 𝒫 A)
32eleq1d 2103 . 2 (z = A → (𝒫 z V ↔ 𝒫 A V))
4 df-pw 3353 . . 3 𝒫 z = {yyz}
5 axpow2 3920 . . . . . 6 xy(yzy x)
65bm1.3ii 3869 . . . . 5 xy(y xyz)
7 abeq2 2143 . . . . . 6 (x = {yyz} ↔ y(y xyz))
87exbii 1493 . . . . 5 (x x = {yyz} ↔ xy(y xyz))
96, 8mpbir 134 . . . 4 x x = {yyz}
109issetri 2558 . . 3 {yyz} V
114, 10eqeltri 2107 . 2 𝒫 z V
121, 3, 11vtocl 2602 1 𝒫 A V
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  Vcvv 2551  wss 2911  𝒫 cpw 3351
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by:  pwexg  3924  p0ex  3930  pp0ex  3931  ord3ex  3932  abexssex  5694  npex  6455  axcnex  6705  pnfxr  8422  mnfxr  8424  ixxex  8498
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