Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  vpwex Unicode version

Theorem vpwex 4103
 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 4104 from vpwex 4103. (Revised by BJ, 10-Aug-2022.)
Assertion
Ref Expression
vpwex

Proof of Theorem vpwex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pw 3512 . 2
2 axpow2 4100 . . . . 5
32bm1.3ii 4049 . . . 4
4 abeq2 2248 . . . . 5
54exbii 1584 . . . 4
63, 5mpbir 145 . . 3
76issetri 2695 . 2
81, 7eqeltri 2212 1
 Colors of variables: wff set class Syntax hints:   wb 104  wal 1329   wceq 1331  wex 1468   wcel 1480  cab 2125  cvv 2686   wss 3071  cpw 3510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512 This theorem is referenced by:  pwexg  4104  pwnex  4370  istopon  12194  dmtopon  12204  tgdom  12255
 Copyright terms: Public domain W3C validator