Theorem snprc 3457

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 3415	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1536	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 626	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3266	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1428	. . 3 |- ( A. x -. x e. { A } <-> -. E. x x e. { A } )
6	4, 5	bitri 182	. 2 |- ( { A } = (/) <-> -. E. x x e. { A } )
7		isset 2605	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 626	. 2 |- ( -. A e. _V <-> -. E. x x = A )
9	3, 6, 8	3bitr4ri 211	1 |- ( -. A e. _V <-> { A } = (/) )

Syntax hints:   -. wn 3   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   (/)c0 3251   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-nul 3252  df-sn 3404

This theorem is referenced by:  prprc1  3500  prprc  3502  snexprc  3958  sucprc  4167  snnen2oprc  6346  unsnfidcex  6385