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Theorem snexALT 4349
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 4365, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4369. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3934 . . 3  |-  { A }  C_  ~P A
2 ssexg 4313 . . 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 4347 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3835 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 187 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4303 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2496 . . 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 1649    e. wcel 1721   _Vcvv 2920    C_ wss 3284   (/)c0 3592   ~Pcpw 3763   {csn 3778
This theorem is referenced by:  p0exALT  4351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298  df-nul 3593  df-pw 3765  df-sn 3784
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