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Theorem snexALT 4134
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 4150, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4154. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3725 . . 3  |-  { A }  C_  ~P A
2 ssexg 4100 . . 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 4132 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3636 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 188 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4090 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2344 . . 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 2740    C_ wss 3094   (/)c0 3397   ~Pcpw 3566   {csn 3581
This theorem is referenced by:  p0exALT  4136  dfiota3  23802  brsuccf  23820  funpartfun  23821  funpartfv  23823  smbkle  25375  cndpv  25381  pgapspf  25384  lineval222  25411  lineval3a  25415  sgplpte21  25464  sgplpte22  25470  isray2  25485  isray  25486
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2742  df-dif 3097  df-in 3101  df-ss 3108  df-nul 3398  df-pw 3568  df-sn 3587
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