Theorem snexprc 4216

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 3684	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 120	. 2 |- ( -. A e. _V -> { A } = (/) )
3		0ex 4157	. 2 |- (/) e. _V
4	2, 3	eqeltrdi 2284	1 |- ( -. A e. _V -> { A } e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   (/)c0 3447   {csn 3619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4156

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-nul 3448  df-sn 3625

This theorem is referenced by:  notnotsnex  4217