Theorem axpow3 4156

Description: A variant of the Axiom of Power Sets ax-pow 4153. 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

Step	Hyp	Ref	Expression
1		axpow2 4155	. . 3 |- E. y A. z ( z C_ x -> z e. y )
2	1	bm1.3ii 4103	. 2 |- E. y A. z ( z e. y <-> z C_ x )
3		bicom 139	. . . 4 |- ( ( z C_ x <-> z e. y ) <-> ( z e. y <-> z C_ x ) )
4	3	albii 1458	. . 3 |- ( A. z ( z C_ x <-> z e. y ) <-> A. z ( z e. y <-> z C_ x ) )
5	4	exbii 1593	. 2 |- ( E. y A. z ( z C_ x <-> z e. y ) <-> E. y A. z ( z e. y <-> z C_ x ) )
6	2, 5	mpbir 145	1 |- E. y A. z ( z C_ x <-> z e. y )

Syntax hints:   <-> wb 104   A.wal 1341   E.wex 1480   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by: (None)