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Theorem vpwex 4197
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 4198 from vpwex 4197. (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 3592 . 2 𝒫 𝑥 = {𝑦𝑦𝑥}
2 axpow2 4194 . . . . 5 𝑧𝑦(𝑦𝑥𝑦𝑧)
32bm1.3ii 4139 . . . 4 𝑧𝑦(𝑦𝑧𝑦𝑥)
4 abeq2 2298 . . . . 5 (𝑧 = {𝑦𝑦𝑥} ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
54exbii 1616 . . . 4 (∃𝑧 𝑧 = {𝑦𝑦𝑥} ↔ ∃𝑧𝑦(𝑦𝑧𝑦𝑥))
63, 5mpbir 146 . . 3 𝑧 𝑧 = {𝑦𝑦𝑥}
76issetri 2761 . 2 {𝑦𝑦𝑥} ∈ V
81, 7eqeltri 2262 1 𝒫 𝑥 ∈ V
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  Vcvv 2752  wss 3144  𝒫 cpw 3590
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by:  pwexg  4198  pwnex  4467  exmidpw2en  6940  istopon  13970  dmtopon  13980  tgdom  14029
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