Theorem snexprc 4188

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 3659	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 120	. 2 |- ( -. A e. _V -> { A } = (/) )
3		0ex 4132	. 2 |- (/) e. _V
4	2, 3	eqeltrdi 2268	1 |- ( -. A e. _V -> { A } e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   (/)c0 3424   {csn 3594

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-nul 3425  df-sn 3600

This theorem is referenced by:  notnotsnex  4189