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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  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 4203 . . 3  |-  { A }  C_  ~P A
2 ssexg 4598 . . 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 4636 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 135 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 4096 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4582 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2563 . . 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 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ~Pcpw 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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