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Theorem pwex 3932
 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 3362 . . 3 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32eleq1d 2106 . 2 (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V))
4 df-pw 3361 . . 3 𝒫 𝑧 = {𝑦𝑦𝑧}
5 axpow2 3929 . . . . . 6 𝑥𝑦(𝑦𝑧𝑦𝑥)
65bm1.3ii 3878 . . . . 5 𝑥𝑦(𝑦𝑥𝑦𝑧)
7 abeq2 2146 . . . . . 6 (𝑥 = {𝑦𝑦𝑧} ↔ ∀𝑦(𝑦𝑥𝑦𝑧))
87exbii 1496 . . . . 5 (∃𝑥 𝑥 = {𝑦𝑦𝑧} ↔ ∃𝑥𝑦(𝑦𝑥𝑦𝑧))
96, 8mpbir 134 . . . 4 𝑥 𝑥 = {𝑦𝑦𝑧}
109issetri 2564 . . 3 {𝑦𝑦𝑧} ∈ V
114, 10eqeltri 2110 . 2 𝒫 𝑧 ∈ V
121, 3, 11vtocl 2608 1 𝒫 𝐴 ∈ V
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361 This theorem is referenced by:  pwexg  3933  p0ex  3939  pp0ex  3940  ord3ex  3941  abexssex  5752  npex  6571  axcnex  6935  pnfxr  8692  mnfxr  8694  ixxex  8768
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