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Theorem axpow3 4207
Description: A variant of the Axiom of Power Sets ax-pow 4204. 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 4206 . . 3  |-  E. y A. z ( z  C_  x  ->  z  e.  y )
21bm1.3ii 4160 . 2  |-  E. y A. z ( z  e.  y  <->  z  C_  x
)
3 bicom 191 . . . 4  |-  ( ( z  C_  x  <->  z  e.  y )  <->  ( z  e.  y  <->  z  C_  x
) )
43albii 1556 . . 3  |-  ( A. z ( z  C_  x 
<->  z  e.  y )  <->  A. z ( z  e.  y  <->  z  C_  x
) )
54exbii 1572 . 2  |-  ( E. y A. z ( z  C_  x  <->  z  e.  y )  <->  E. y A. z ( z  e.  y  <->  z  C_  x
) )
62, 5mpbir 200 1  |-  E. y A. z ( z  C_  x 
<->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530   E.wex 1531    e. wcel 1696    C_ wss 3165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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