Theorem snexg 3956

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

Step	Hyp	Ref	Expression
1		pwexg 3954	. 2 |- ( A e. V -> ~P A e. _V )
2		snsspw 3556	. . 3 |- { A } C_ ~P A
3		ssexg 3917	. . 3 |- ( ( { A } C_ ~P A /\ ~P A e. _V ) -> { A } e. _V )
4	2, 3	mpan 414	. 2 |- ( ~P A e. _V -> { A } e. _V )
5	1, 4	syl 14	1 |- ( A e. V -> { A } e. _V )

Syntax hints:   -> wi 4   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382   {csn 3398

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by:  snex  3957  snelpwi  3967  opexg  3983  opm  3989  tpexg  4197  op1stbg  4228  sucexb  4241  elxp4  4828  elxp5  4829  opabex3d  5768  opabex3  5769  1stvalg  5789  2ndvalg  5790  mpt2exxg  5853  cnvf1o  5866  brtpos2  5889  tfr0  5960  tfrlemisucaccv  5962  tfrlemibxssdm  5964  tfrlemibfn  5965  xpsnen2g  6326  iseqid3  9465  climconst2  10130