Theorem elpwuni 3866

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 3861	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3729	. . . 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 3115	. . 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 1312   e. wcel 1461   C_ wss 3035   ~Pcpw 3474   U.cuni 3700

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-uni 3701

This theorem is referenced by: (None)