Theorem axpow3 3930

Description: A variant of the Axiom of Power Sets ax-pow 3927. 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 3929	. . 3 |- E. y A. z ( z C_ x -> z e. y )
2	1	bm1.3ii 3878	. 2 |- E. y A. z ( z e. y <-> z C_ x )
3		bicom 128	. . . 4 |- ( ( z C_ x <-> z e. y ) <-> ( z e. y <-> z C_ x ) )
4	3	albii 1359	. . 3 |- ( A. z ( z C_ x <-> z e. y ) <-> A. z ( z e. y <-> z C_ x ) )
5	4	exbii 1496	. 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 134	1 |- E. y A. z ( z C_ x <-> z e. y )

Syntax hints:   <-> wb 98   A.wal 1241   E.wex 1381   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by: (None)