Theorem snexprc 4321

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 3773	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 120	. 2 |- ( -. A e. _V -> { A } = (/) )
3		0ex 4258	. 2 |- (/) e. _V
4	2, 3	eqeltrdi 2329	1 |- ( -. A e. _V -> { A } e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {csn 3708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4257

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521  df-sn 3714

This theorem is referenced by:  notnotsnex  4322