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Theorem vpwex 5013
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 5014 from vpwex 5013. (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 4317 . 2 𝒫 𝑥 = {𝑦𝑦𝑥}
2 axpow2 5003 . . . . 5 𝑧𝑦(𝑦𝑥𝑦𝑧)
32bm1.3ii 4944 . . . 4 𝑧𝑦(𝑦𝑧𝑦𝑥)
4 abeq2 2875 . . . . 5 (𝑧 = {𝑦𝑦𝑥} ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
54exbii 1943 . . . 4 (∃𝑧 𝑧 = {𝑦𝑦𝑥} ↔ ∃𝑧𝑦(𝑦𝑧𝑦𝑥))
63, 5mpbir 222 . . 3 𝑧 𝑧 = {𝑦𝑦𝑥}
76issetri 3363 . 2 {𝑦𝑦𝑥} ∈ V
81, 7eqeltri 2840 1 𝒫 𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  Vcvv 3350  wss 3732  𝒫 cpw 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743  ax-sep 4941  ax-pow 5001
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352  df-in 3739  df-ss 3746  df-pw 4317
This theorem is referenced by:  pwexg  5014  pwnex  7166  inf3lem7  8746  dfac8  9210  dfac13  9217  ackbij1lem8  9302  dominf  9520  numthcor  9569  dominfac  9648  intwun  9810  wunex2  9813  eltsk2g  9826  inttsk  9849  tskcard  9856  intgru  9889  gruina  9893  axgroth6  9903  ismre  16516  fnmre  16517  mreacs  16584  isacs5lem  17435  pmtrfval  18133  istopon  20996  dmtopon  21007  tgdom  21062  isfbas  21912  bj-snglex  33388  pwinfi  38544  ntrrn  39094  ntrf  39095  dssmapntrcls  39100
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