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Theorem vpwex 4179
Description: Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4180 from vpwex 4179. (Revised by BJ, 10-Aug-2022.)
Assertion
Ref Expression
vpwex 𝒫 𝑥 ∈ V

Proof of Theorem vpwex
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pw 3577 . 2 𝒫 𝑥 = {𝑦𝑦𝑥}
2 axpow2 4176 . . . . 5 𝑧𝑦(𝑦𝑥𝑦𝑧)
32bm1.3ii 4124 . . . 4 𝑧𝑦(𝑦𝑧𝑦𝑥)
4 abeq2 2286 . . . . 5 (𝑧 = {𝑦𝑦𝑥} ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
54exbii 1605 . . . 4 (∃𝑧 𝑧 = {𝑦𝑦𝑥} ↔ ∃𝑧𝑦(𝑦𝑧𝑦𝑥))
63, 5mpbir 146 . . 3 𝑧 𝑧 = {𝑦𝑦𝑥}
76issetri 2746 . 2 {𝑦𝑦𝑥} ∈ V
81, 7eqeltri 2250 1 𝒫 𝑥 ∈ V
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  Vcvv 2737  wss 3129  𝒫 cpw 3575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577
This theorem is referenced by:  pwexg  4180  pwnex  4449  istopon  13483  dmtopon  13493  tgdom  13542
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