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Theorem snexg 3910
Description: A singleton whose element exists is a set. The  _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  V  { }  _V

Proof of Theorem snexg
StepHypRef Expression
1 pwexg 3907 . 2  V  ~P  _V
2 snsspw 3509 . . 3  { }  C_  ~P
3 ssexg 3870 . . 3  { }  C_  ~P  ~P  _V  { }  _V
42, 3mpan 402 . 2  ~P  _V  { }  _V
51, 4syl 14 1  V  { }  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1374   _Vcvv 2535    C_ wss 2894   ~Pcpw 3334   {csn 3350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356
This theorem is referenced by:  snex  3911  opexg  3938  tpexg  4129  opabex3d  5671  opabex3  5672  mpt2exxg  5756  cnvf1o  5769  brtpos2  5788  tfrlemisucaccv  5860  tfrlemibxssdm  5862  tfrlemibfn  5863
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