Theorem pwel 4247

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 3863	. . 3 |- ( A e. B -> A C_ U. B )
2		sspwb 4245	. . 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 4209	. . 3 |- ( A e. B -> ~P A e. _V )
5		elpwg 3609	. . 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 2164   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   U.cuni 3835

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-uni 3836

This theorem is referenced by: (None)