Theorem snprc 3435

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 3392	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1496	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 594	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3239	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1388	. . 3 |- ( A. x -. x e. { A } <-> -. E. x x e. { A } )
6	4, 5	bitri 173	. 2 |- ( { A } = (/) <-> -. E. x x e. { A } )
7		isset 2561	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 594	. 2 |- ( -. A e. _V <-> -. E. x x = A )
9	3, 6, 8	3bitr4ri 202	1 |- ( -. A e. _V <-> { A } = (/) )

Syntax hints:   -. wn 3   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   (/)c0 3224   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-sn 3381

This theorem is referenced by:  prprc1  3478  prprc  3480  snexprc  3938  sucprc  4149  snnen2oprc  6323