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Theorem snexg 4010
Description: A singleton whose element exists is a set. The  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
Assertion
Ref Expression
snexg  |-  ( A  e.  V  ->  { A }  e.  _V )

Proof of Theorem snexg
StepHypRef Expression
1 pwexg 4007 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 snsspw 3603 . . 3  |-  { A }  C_  ~P A
3 ssexg 3970 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
42, 3mpan 415 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
51, 4syl 14 1  |-  ( A  e.  V  ->  { A }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425   {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447
This theorem is referenced by:  snex  4011  notnotsnex  4013  snelpwi  4030  opexg  4046  opm  4052  tpexg  4260  op1stbg  4291  sucexb  4304  elxp4  4905  elxp5  4906  opabex3d  5874  opabex3  5875  1stvalg  5895  2ndvalg  5896  mpt2exxg  5959  cnvf1o  5972  brtpos2  5998  tfr0dm  6069  tfrlemisucaccv  6072  tfrlemibxssdm  6074  tfrlemibfn  6075  tfr1onlemsucaccv  6088  tfr1onlembxssdm  6090  tfr1onlembfn  6091  tfrcllemsucaccv  6101  tfrcllembxssdm  6103  tfrcllembfn  6104  fvdiagfn  6430  xpsnen2g  6525  zfz1isolem1  10210  climconst2  10643
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