Theorem elpwuni 4006

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 4001	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3868	. . . 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 3237	. . 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 1364   e. wcel 2167   C_ wss 3157   ~Pcpw 3605   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840

This theorem is referenced by: (None)