Theorem pwel 4262

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 3878	. . 3 |- ( A e. B -> A C_ U. B )
2		sspwb 4260	. . 3 |- ( A C_ U. B <-> ~P A C_ ~P U. B )
3	1, 2	sylib 122	. 2 |- ( A e. B -> ~P A C_ ~P U. B )
4		pwexg 4224	. . 3 |- ( A e. B -> ~P A e. _V )
5		elpwg 3624	. . 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 167	1 |- ( A e. B -> ~P A e. ~P ~P U. B )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   U.cuni 3850

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-uni 3851

This theorem is referenced by: (None)