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Theorem snexALT 4377
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 4393, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4397. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3962 . . 3  |-  { A }  C_  ~P A
2 ssexg 4341 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
31, 2mpan 652 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
4 pwexg 4375 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3863 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 187 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4331 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2523 . . 3  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
105, 9syl 16 . 2  |-  ( -. 
~P A  e.  _V  ->  { A }  e.  _V )
113, 10pm2.61i 158 1  |-  { A }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806
This theorem is referenced by:  p0exALT  4379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812
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