Theorem elpwuni 3741

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 3739	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissel 3609	. . . 4 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
3	2	expcom 109	. . 3 |- ( B e. A -> ( U. A C_ B -> U. A = B ) )
4		eqimss 2997	. . 3 |- ( U. A = B -> U. A C_ B )
5	3, 4	impbid1 130	. 2 |- ( B e. A -> ( U. A C_ B <-> U. A = B ) )
6	1, 5	syl5bb 181	1 |- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   C_ wss 2917   ~Pcpw 3359   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581

This theorem is referenced by: (None)