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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
StepHypRef Expression
1 elex 3118 . 2
2 elex 3118 . . 3
3 eldifpw.1 . . . 4
4 difex2 6607 . . . 4
53, 4ax-mp 5 . . 3
62, 5sylibr 212 . 2
7 elpwg 4020 . . 3
8 difexg 4600 . . . . 5
9 elpwg 4020 . . . . 5
108, 9syl 16 . . . 4
11 uncom 3647 . . . . . 6
1211sseq2i 3528 . . . . 5
13 ssundif 3911 . . . . 5
1412, 13bitri 249 . . . 4
1510, 14syl6rbbr 264 . . 3
167, 15bitrd 253 . 2
171, 6, 16pm5.21nii 353 1
 Colors of variables: wff setvar class 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
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