Theorem pwel 3973

Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)

Assertion

Ref	Expression
pwel	|- ( A e. B -> ~P A e. ~P ~P U. B )

Proof of Theorem pwel

Step	Hyp	Ref	Expression
1		elssuni 3629	. . 3 |- ( A e. B -> A C_ U. B )
2		sspwb 3971	. . 3 |- ( A C_ U. B <-> ~P A C_ ~P U. B )
3	1, 2	sylib 120	. 2 |- ( A e. B -> ~P A C_ ~P U. B )
4		pwexg 3954	. . 3 |- ( A e. B -> ~P A e. _V )
5		elpwg 3390	. . 3 |- ( ~P A e. _V -> ( ~P A e. ~P ~P U. B <-> ~P A C_ ~P U. B ) )
6	4, 5	syl 14	. 2 |- ( A e. B -> ( ~P A e. ~P ~P U. B <-> ~P A C_ ~P U. B ) )
7	3, 6	mpbird 165	1 |- ( A e. B -> ~P A e. ~P ~P U. B )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-uni 3602

This theorem is referenced by: (None)