MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpow3 Structured version   Visualization version   GIF version

Theorem axpow3 5232
Description: A variant of the Axiom of Power Sets ax-pow 5229. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow3 𝑦𝑧(𝑧𝑥𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axpow3
StepHypRef Expression
1 axpow2 5231 . . 3 𝑦𝑧(𝑧𝑥𝑧𝑦)
21bm1.3ii 5167 . 2 𝑦𝑧(𝑧𝑦𝑧𝑥)
3 bicom 225 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑦𝑧𝑥))
43albii 1826 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(𝑧𝑦𝑧𝑥))
54exbii 1854 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(𝑧𝑦𝑧𝑥))
62, 5mpbir 234 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1540  wex 1786  wss 3841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5164  ax-pow 5229
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-in 3848  df-ss 3858
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator