Theorem elpwuni 4616

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 4611	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 4468	. . . 4 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
3	2	expcom 451	. . 3 |- ( B e. A -> ( U. A C_ B -> U. A = B ) )
4		eqimss 3657	. . 3 |- ( U. A = B -> U. A C_ B )
5	3, 4	impbid1 215	. 2 |- ( B e. A -> ( U. A C_ B <-> U. A = B ) )
6	1, 5	syl5bb 272	1 |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  mreuni  16260  ustuni  22030  utopbas  22039  issgon  30186  br2base  30331