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Theorem snexALT 4198
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 4214, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4218. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3786 . . 3  |-  { A }  C_  ~P A
2 ssexg 4162 . . 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 4196 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 127 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3697 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 186 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4152 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2373 . . 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 1625    e. wcel 1686   _Vcvv 2790    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   {csn 3642
This theorem is referenced by:  p0exALT  4200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-in 3161  df-ss 3168  df-nul 3458  df-pw 3629  df-sn 3648
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