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Theorem elpwunsn 4074
 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 3491 . 2
2 elpwg 4024 . . . . . . 7
3 dfss3 3499 . . . . . . 7
42, 3syl6bb 261 . . . . . 6
54notbid 294 . . . . 5
65biimpa 484 . . . 4
7 rexnal 2915 . . . 4
86, 7sylibr 212 . . 3
9 elpwi 4025 . . . . . . . . . 10
10 ssel 3503 . . . . . . . . . 10
11 elun 3650 . . . . . . . . . . . . 13
12 elsni 4058 . . . . . . . . . . . . . . 15
1312orim2i 518 . . . . . . . . . . . . . 14
1413ord 377 . . . . . . . . . . . . 13
1511, 14sylbi 195 . . . . . . . . . . . 12
1615imim2i 14 . . . . . . . . . . 11
1716impd 431 . . . . . . . . . 10
189, 10, 173syl 20 . . . . . . . . 9
19 eleq1 2539 . . . . . . . . . 10
2019biimpd 207 . . . . . . . . 9
2118, 20syl6 33 . . . . . . . 8
2221expd 436 . . . . . . 7
2322com4r 86 . . . . . 6
2423pm2.43b 50 . . . . 5
2524rexlimdv 2957 . . . 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 1379   wcel 1767  wral 2817  wrex 2818   cdif 3478   cun 3479   wss 3481  cpw 4016  csn 4033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pw 4018  df-sn 4034 This theorem is referenced by:  pwfilem  7826  incexclem  13628  ramub1lem1  14420  ptcmplem5  20424  onsucsuccmpi  29835
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