Theorem elpwuni 4054

Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
elpwuni	|- (B e. A -> (A C_ ~PB <-> U.A = B))

Proof of Theorem elpwuni

Step	Hyp	Ref	Expression
1		sspwuni 4052	. 2 |- (A C_ ~PB <-> U.A C_ B)
2		unissel 3921	. . . 4 |- ((U.A C_ B /\ B e. A) -> U.A = B)
3	2	expcom 424	. . 3 |- (B e. A -> (U.A C_ B -> U.A = B))
4		eqimss 3324	. . 3 |- (U.A = B -> U.A C_ B)
5	3, 4	impbid1 194	. 2 |- (B e. A -> (U.A C_ B <-> U.A = B))
6	1, 5	syl5bb 248	1 |- (B e. A -> (A C_ ~PB <-> U.A = B))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710   C_ wss 3258  ~Pcpw 3723  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725  df-uni 3893

This theorem is referenced by: (None)