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Theorem snexALT 4196
Description: A singleton is a set. Theorem 7.13 of [Quine] p. 51, but proved using only Extensionality, Power Set, and Separation. Unlike the proof of zfpair 4212, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4216. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3784 . . 3  |-  { A }  C_  ~P A
2 ssexg 4160 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
31, 2mpan 651 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
4 pwexg 4194 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 127 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3695 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 186 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4150 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2371 . . 3  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
105, 9syl 15 . 2  |-  ( -. 
~P A  e.  _V  ->  { A }  e.  _V )
113, 10pm2.61i 156 1  |-  { A }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640
This theorem is referenced by:  p0exALT  4198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646
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