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

Theorem vpwex 4071
 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 4072 from vpwex 4071. (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 3480 . 2 𝒫 𝑥 = {𝑦𝑦𝑥}
2 axpow2 4068 . . . . 5 𝑧𝑦(𝑦𝑥𝑦𝑧)
32bm1.3ii 4017 . . . 4 𝑧𝑦(𝑦𝑧𝑦𝑥)
4 abeq2 2224 . . . . 5 (𝑧 = {𝑦𝑦𝑥} ↔ ∀𝑦(𝑦𝑧𝑦𝑥))
54exbii 1567 . . . 4 (∃𝑧 𝑧 = {𝑦𝑦𝑥} ↔ ∃𝑧𝑦(𝑦𝑧𝑦𝑥))
63, 5mpbir 145 . . 3 𝑧 𝑧 = {𝑦𝑦𝑥}
76issetri 2667 . 2 {𝑦𝑦𝑥} ∈ V
81, 7eqeltri 2188 1 𝒫 𝑥 ∈ V
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∀wal 1312   = wceq 1314  ∃wex 1451   ∈ wcel 1463  {cab 2101  Vcvv 2658   ⊆ wss 3039  𝒫 cpw 3478 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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066 This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480 This theorem is referenced by:  pwexg  4072  pwnex  4338  istopon  12086  dmtopon  12096  tgdom  12147
 Copyright terms: Public domain W3C validator