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Theorem snexALT 4477
Description: Alternate proof of snex 4532 using Power Set (ax-pow 4469) instead of Pairing (ax-pr 4530). Unlike in the proof of zfpair 4528, Replacement (ax-rep 4402) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4532. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 4043 . . 3  |-  { A }  C_  ~P A
2 ssexg 4437 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
31, 2mpan 670 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
4 pwexg 4475 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 135 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3938 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4421 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2530 . . 3  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
105, 9syl 16 . 2  |-  ( -. 
~P A  e.  _V  ->  { A }  e.  _V )
113, 10pm2.61i 164 1  |-  { A }  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2971    C_ wss 3327   (/)c0 3636   ~Pcpw 3859   {csn 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2973  df-dif 3330  df-in 3334  df-ss 3341  df-nul 3637  df-pw 3861  df-sn 3877
This theorem is referenced by:  p0exALT  4479
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