Theorem snexg 4163

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 4159	. 2 |- ( A e. V -> ~P A e. _V )
2		snsspw 3744	. . 3 |- { A } C_ ~P A
3		ssexg 4121	. . 3 |- ( ( { A } C_ ~P A /\ ~P A e. _V ) -> { A } e. _V )
4	2, 3	mpan 421	. 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 2136   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582

This theorem is referenced by:  snex  4164  notnotsnex  4166  exmidsssnc  4182  snelpwi  4190  opexg  4206  opm  4212  tpexg  4422  op1stbg  4457  sucexb  4474  elxp4  5091  elxp5  5092  opabex3d  6089  opabex3  6090  1stvalg  6110  2ndvalg  6111  mpoexxg  6178  cnvf1o  6193  brtpos2  6219  tfr0dm  6290  tfrlemisucaccv  6293  tfrlemibxssdm  6295  tfrlemibfn  6296  tfr1onlemsucaccv  6309  tfr1onlembxssdm  6311  tfr1onlembfn  6312  tfrcllemsucaccv  6322  tfrcllembxssdm  6324  tfrcllembfn  6325  fvdiagfn  6659  ixpsnf1o  6702  mapsnf1o  6703  xpsnen2g  6795  zfz1isolem1  10753  climconst2  11232  ennnfonelemp1  12339  setsvalg  12424  setsex  12426  setsslid  12444  strle1g  12485  1strbas  12494  mgm1  12601