Theorem snprc 3708

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 3660	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1629	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 670	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3487	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1523	. . 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 2783	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 670	. 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 1371   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   (/)c0 3468   {csn 3643

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-nul 3469  df-sn 3649

This theorem is referenced by:  prprc1  3751  prprc  3753  snexprc  4246  sucprc  4477  snnen2oprc  6982  unsnfidcex  7043