Theorem elpwuni 4418

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

Assertion

Ref	Expression
elpwuni

Proof of Theorem elpwuni

Step	Hyp	Ref	Expression
1		sspwuni 4416	. 2
2		unissel 4280	. . . 4
3	2	expcom 435	. . 3
4		eqimss 3555	. . 3
5	3, 4	impbid1 203	. 2
6	1, 5	syl5bb 257	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  C_wss 3475  ~Pcpw 4012  U.cuni 4249

This theorem is referenced by:  mreuni  14997  ustuni  20729  utopbas  20738  issgon  28123  br2base  28240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014  df-uni 4250