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

Theorem axpow3 4191
Description: A variant of the Axiom of Power Sets ax-pow 4188. 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  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Distinct variable group:    x, y, z

Proof of Theorem axpow3
StepHypRef Expression
1 axpow2 4190 . . 3  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
21bm1.3ii 4146 . 2  |-  E. y A. z ( z  e.  y  <->  z  C_  x
)
3 bicom 193 . . . 4  |-  ( ( z  C_  x  <->  z  e.  y )  <->  ( z  e.  y  <->  z  C_  x
) )
43albii 1554 . . 3  |-  ( A. z ( z  C_  x 
<->  z  e.  y )  <->  A. z ( z  e.  y  <->  z  C_  x
) )
54exbii 1570 . 2  |-  ( E. y A. z ( z  C_  x  <->  z  e.  y )  <->  E. y A. z ( z  e.  y  <->  z  C_  x
) )
62, 5mpbir 202 1  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1528   E.wex 1529    e. wcel 1685    C_ wss 3154
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-pow 4188
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-in 3161  df-ss 3168
  Copyright terms: Public domain W3C validator