Theorem snprc 3734

Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
snprc	|- ( -. A e. _V <-> { A } = (/) )

Proof of Theorem snprc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3686	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1653	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 674	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3513	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1547	. . 3 |- ( A. x -. x e. { A } <-> -. E. x x e. { A } )
6	4, 5	bitri 184	. 2 |- ( { A } = (/) <-> -. E. x x e. { A } )
7		isset 2809	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 674	. 2 |- ( -. A e. _V <-> -. E. x x = A )
9	3, 6, 8	3bitr4ri 213	1 |- ( -. A e. _V <-> { A } = (/) )

Syntax hints:   -. wn 3   <-> wb 105   A.wal 1395   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   (/)c0 3494   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495  df-sn 3675

This theorem is referenced by:  prprc1  3780  prprc  3782  snexprc  4276  sucprc  4509  snnen2oprc  7045  unsnfidcex  7111