Theorem snprc 3731

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 3683	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1651	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 672	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3510	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1545	. . 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 2806	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 672	. 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 1393   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   (/)c0 3491   {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by:  prprc1  3775  prprc  3777  snexprc  4270  sucprc  4503  snnen2oprc  7021  unsnfidcex  7082