Theorem pwel 4140

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 3764	. . 3 |- ( A e. B -> A C_ U. B )
2		sspwb 4138	. . 3 |- ( A C_ U. B <-> ~P A C_ ~P U. B )
3	1, 2	sylib 121	. 2 |- ( A e. B -> ~P A C_ ~P U. B )
4		pwexg 4104	. . 3 |- ( A e. B -> ~P A e. _V )
5		elpwg 3518	. . 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 166	1 |- ( A e. B -> ~P A e. ~P ~P U. B )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   _Vcvv 2686   C_ wss 3071   ~Pcpw 3510   U.cuni 3736

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3737

This theorem is referenced by: (None)