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Theorem vpwex 4006
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 4007 from vpwex 4006. (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 3427 . 2 𝒫 𝑥 = {𝑦𝑦𝑥}
2 axpow2 4003 . . . . 5 𝑧𝑦(𝑦𝑥𝑦𝑧)
32bm1.3ii 3952 . . . 4 𝑧𝑦(𝑦𝑧𝑦𝑥)
4 abeq2 2196 . . . . 5 (𝑧 = {𝑦𝑦𝑥} ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
54exbii 1541 . . . 4 (∃𝑧 𝑧 = {𝑦𝑦𝑥} ↔ ∃𝑧𝑦(𝑦𝑧𝑦𝑥))
63, 5mpbir 144 . . 3 𝑧 𝑧 = {𝑦𝑦𝑥}
76issetri 2628 . 2 {𝑦𝑦𝑥} ∈ V
81, 7eqeltri 2160 1 𝒫 𝑥 ∈ V
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  {cab 2074  Vcvv 2619  wss 2997  𝒫 cpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pwexg  4007
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