Theorem snexprc 3958

Description: A singleton whose element is a proper class 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
snexprc	|- ( -. A e. _V -> { A } e. _V )

Proof of Theorem snexprc

Step	Hyp	Ref	Expression
1		snprc 3457	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 118	. 2 |- ( -. A e. _V -> { A } = (/) )
3		0ex 3905	. 2 |- (/) e. _V
4	2, 3	syl6eqel 2169	1 |- ( -. A e. _V -> { A } e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   (/)c0 3251   {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-in1 576  ax-in2 577  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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252  df-sn 3404

This theorem is referenced by: (None)