Theorem elpwun 6613

Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)

Hypothesis

Ref	Expression
eldifpw.1

Assertion

Ref	Expression
elpwun

Proof of Theorem elpwun

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		elex 3118	. . 3
3		eldifpw.1	. . . 4
4		difex2 6607	. . . 4
5	3, 4	ax-mp 5	. . 3
6	2, 5	sylibr 212	. 2
7		elpwg 4020	. . 3
8		difexg 4600	. . . . 5
9		elpwg 4020	. . . . 5
10	8, 9	syl 16	. . . 4
11		uncom 3647	. . . . . 6
12	11	sseq2i 3528	. . . . 5
13		ssundif 3911	. . . . 5
14	12, 13	bitri 249	. . . 4
15	10, 14	syl6rbbr 264	. . 3
16	7, 15	bitrd 253	. 2
17	1, 6, 16	pm5.21nii 353	1

Syntax hints:  <->wb 184  e.wcel 1818  cvv 3109  \cdif 3472  u.cun 3473  C_wss 3475  ~Pcpw 4012

This theorem is referenced by:  pwfilem  7834  elrfi  30626

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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  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-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-pw 4014  df-sn 4030  df-pr 4032  df-uni 4250