Theorem snprc 3770

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 3722	. . . 4 |- ( x e. { A } <-> x = A )
2	1	exbii 1658	. . 3 |- ( E. x x e. { A } <-> E. x x = A )
3	2	notbii 678	. 2 |- ( -. E. x x e. { A } <-> -. E. x x = A )
4		eq0 3540	. . 3 |- ( { A } = (/) <-> A. x -. x e. { A } )
5		alnex 1552	. . 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 2828	. . 3 |- ( A e. _V <-> E. x x = A )
8	7	notbii 678	. 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 1400   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {csn 3705

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521  df-sn 3711

This theorem is referenced by:  prprc1  3816  prprc  3818  snexprc  4318  sucprc  4552  snnen2oprc  7151  unsnfidcex  7217