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Theorem snexALT 4638
 Description: Alternate proof of snex 4693 using Power Set (ax-pow 4630) instead of Pairing (ax-pr 4691). Unlike in the proof of zfpair 4689, Replacement (ax-rep 4563) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4693. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 4203 . . 3
2 ssexg 4598 . . 3
31, 2mpan 670 . 2
4 pwexg 4636 . . . 4
54con3i 135 . . 3
6 snprc 4096 . . . . 5
76biimpi 194 . . . 4
8 0ex 4582 . . . 4
97, 8syl6eqel 2563 . . 3
105, 9syl 16 . 2
113, 10pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1379   wcel 1767  cvv 3118   wss 3481  c0 3790  cpw 4015  csn 4032 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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630 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-ne 2664  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4017  df-sn 4033 This theorem is referenced by:  p0exALT  4640
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