Theorem pwel 4310

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 3921	. . 3 |- ( A e. B -> A C_ U. B )
2		sspwb 4308	. . 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 4270	. . 3 |- ( A e. B -> ~P A e. _V )
5		elpwg 3660	. . 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 2202   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   U.cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-uni 3894

This theorem is referenced by: (None)