Theorem axpow3 4172

Description: A variant of the Axiom of Power Sets ax-pow 4169. 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 4171	. . 3 |- E. y A. z ( z C_ x -> z e. y )
2	1	bm1.3ii 4119	. 2 |- E. y A. z ( z e. y <-> z C_ x )
3		bicom 140	. . . 4 |- ( ( z C_ x <-> z e. y ) <-> ( z e. y <-> z C_ x ) )
4	3	albii 1468	. . 3 |- ( A. z ( z C_ x <-> z e. y ) <-> A. z ( z e. y <-> z C_ x ) )
5	4	exbii 1603	. 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 146	1 |- E. y A. z ( z C_ x <-> z e. y )

Syntax hints:   <-> wb 105   A.wal 1351   E.wex 1490   C_ wss 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-in 3133  df-ss 3140

This theorem is referenced by: (None)