Theorem snprc 3756

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 3708	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1654	. . 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 3529	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1548	. . 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 2822	. . 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 1396   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   (/)c0 3510   {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3215  df-nul 3511  df-sn 3697

This theorem is referenced by:  prprc1  3802  prprc  3804  snexprc  4301  sucprc  4535  snnen2oprc  7116  unsnfidcex  7182