Theorem snexprc 4050

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 3535	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 119	. 2 |- ( -. A e. _V -> { A } = (/) )
3		0ex 3995	. 2 |- (/) e. _V
4	2, 3	syl6eqel 2190	1 |- ( -. A e. _V -> { A } e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1299   e. wcel 1448   _Vcvv 2641   (/)c0 3310   {csn 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-nul 3311  df-sn 3480

This theorem is referenced by:  notnotsnex  4051