Theorem snprc 3552

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 3508	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1565	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 640	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3345	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1456	. . 3 |- ( A. x -. x e. { A } <-> -. E. x x e. { A } )
6	4, 5	bitri 183	. 2 |- ( { A } = (/) <-> -. E. x x e. { A } )
7		isset 2661	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 640	. 2 |- ( -. A e. _V <-> -. E. x x = A )
9	3, 6, 8	3bitr4ri 212	1 |- ( -. A e. _V <-> { A } = (/) )

Syntax hints:   -. wn 3   <-> wb 104   A.wal 1310   = wceq 1312   E.wex 1449   e. wcel 1461   _Vcvv 2655   (/)c0 3327   {csn 3491

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-nul 3328  df-sn 3497

This theorem is referenced by:  prprc1  3595  prprc  3597  snexprc  4068  sucprc  4292  snnen2oprc  6705  unsnfidcex  6759