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Theorem snexALT 4168
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 4184, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4188. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3758 . . 3  |-  { A }  C_  ~P A
2 ssexg 4134 . . 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 4166 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3669 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 188 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4124 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2346 . . 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 2763    C_ wss 3127   (/)c0 3430   ~Pcpw 3599   {csn 3614
This theorem is referenced by:  p0exALT  4170  dfiota3  23838  brsuccf  23856  funpartfun  23857  funpartfv  23859  smbkle  25411  cndpv  25417  pgapspf  25420  lineval222  25447  lineval3a  25451  sgplpte21  25500  sgplpte22  25506  isray2  25521  isray  25522
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141  df-nul 3431  df-pw 3601  df-sn 3620
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