Theorem elpwuni 3762

Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
elpwuni	|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Proof of Theorem elpwuni

Step	Hyp	Ref	Expression
1		sspwuni 3760	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3630	. . . 4 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
3	2	expcom 114	. . 3 |- ( B e. A -> ( U. A C_ B -> U. A = B ) )
4		eqimss 3051	. . 3 |- ( U. A = B -> U. A C_ B )
5	3, 4	impbid1 140	. 2 |- ( B e. A -> ( U. A C_ B <-> U. A = B ) )
6	1, 5	syl5bb 190	1 |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

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-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-uni 3602

This theorem is referenced by: (None)