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Theorem snexALT 4090
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 4106, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4110. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3684 . . 3  |-  { A }  C_  ~P A
2 ssexg 4057 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
31, 2mpan 654 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
4 pwexg 4088 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3599 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 188 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4047 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2341 . . 3  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
105, 9syl 17 . 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 5    = wceq 1619    e. wcel 1621   _Vcvv 2727    C_ wss 3078   (/)c0 3362   ~Pcpw 3530   {csn 3544
This theorem is referenced by:  p0exALT  4092  dfiota3  23636  brsuccf  23654  funpartfun  23655  funpartfv  23657  smbkle  25209  cndpv  25215  pgapspf  25218  lineval222  25245  lineval3a  25249  sgplpte21  25298  sgplpte22  25304  isray2  25319  isray  25320
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363  df-pw 3532  df-sn 3550
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