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Theorem elpwunsn 4070
 Description: Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
Assertion
Ref Expression
elpwunsn

Proof of Theorem elpwunsn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldif 3485 . 2
2 elpwg 4020 . . . . . . 7
3 dfss3 3493 . . . . . . 7
42, 3syl6bb 261 . . . . . 6
54notbid 294 . . . . 5
65biimpa 484 . . . 4
7 rexnal 2905 . . . 4
86, 7sylibr 212 . . 3
9 elpwi 4021 . . . . . . . . . 10
10 ssel 3497 . . . . . . . . . 10
11 elun 3644 . . . . . . . . . . . . 13
12 elsni 4054 . . . . . . . . . . . . . . 15
1312orim2i 518 . . . . . . . . . . . . . 14
1413ord 377 . . . . . . . . . . . . 13
1511, 14sylbi 195 . . . . . . . . . . . 12
1615imim2i 14 . . . . . . . . . . 11
1716impd 431 . . . . . . . . . 10
189, 10, 173syl 20 . . . . . . . . 9
19 eleq1 2529 . . . . . . . . . 10
2019biimpd 207 . . . . . . . . 9
2118, 20syl6 33 . . . . . . . 8
2221expd 436 . . . . . . 7
2322com4r 86 . . . . . 6
2423pm2.43b 50 . . . . 5
2524rexlimdv 2947 . . . 4
2625imp 429 . . 3
278, 26syldan 470 . 2
281, 27sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  \cdif 3472  u.cun 3473  C_wss 3475  ~Pcpw 4012  {csn 4029 This theorem is referenced by:  pwfilem  7834  incexclem  13648  ramub1lem1  14544  ptcmplem5  20556  onsucsuccmpi  29908 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-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-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pw 4014  df-sn 4030
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