Theorem elpwuni 4065

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 4060	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3927	. . . 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 3282	. . 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 1398   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-uni 3899

This theorem is referenced by: (None)