Theorem elpwuni 3978

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 3973	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3840	. . . 4 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
3	2	expcom 116	. . 3 |- ( B e. A -> ( U. A C_ B -> U. A = B ) )
4		eqimss 3211	. . 3 |- ( U. A = B -> U. A C_ B )
5	3, 4	impbid1 142	. 2 |- ( B e. A -> ( U. A C_ B <-> U. A = B ) )
6	1, 5	bitrid 192	1 |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   C_ wss 3131   ~Pcpw 3577   U.cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-uni 3812

This theorem is referenced by: (None)