Theorem elpwuni 3940

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 3935	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3803	. . . 4 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
3	2	expcom 115	. . 3 |- ( B e. A -> ( U. A C_ B -> U. A = B ) )
4		eqimss 3182	. . 3 |- ( U. A = B -> U. A C_ B )
5	3, 4	impbid1 141	. 2 |- ( B e. A -> ( U. A C_ B <-> U. A = B ) )
6	1, 5	syl5bb 191	1 |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1335   e. wcel 2128   C_ wss 3102   ~Pcpw 3544   U.cuni 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-pw 3546  df-uni 3775

This theorem is referenced by: (None)