Theorem snexg 3936

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 3933	. 2 |- ( A e. V -> ~P A e. _V )
2		snsspw 3535	. . 3 |- { A } C_ ~P A
3		ssexg 3896	. . 3 |- ( ( { A } C_ ~P A /\ ~P A e. _V ) -> { A } e. _V )
4	2, 3	mpan 400	. 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 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359   {csn 3375

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by:  snex  3937  opexg  3964  tpexg  4179  opabex3d  5748  opabex3  5749  mpt2exxg  5833  cnvf1o  5846  brtpos2  5866  tfr0  5937  tfrlemisucaccv  5939  tfrlemibxssdm  5941  tfrlemibfn  5942  xpsnen2g  6303  iseqid3  9245  climconst2  9812