Theorem snprc 3788

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		elsn 3748	. . . 4 |- (x e. {A} <-> x = A)
2	1	exbii 1582	. . 3 |- (E.x x e. {A} <-> E.x x = A)
3		neq0 3560	. . 3 |- (-. {A} = (/) <-> E.x x e. {A})
4		isset 2863	. . 3 |- (A e. _V <-> E.x x = A)
5	2, 3, 4	3bitr4i 268	. 2 |- (-. {A} = (/) <-> A e. _V)
6	5	con1bii 321	1 |- (-. A e. _V <-> {A} = (/))

Syntax hints:  -. wn 3   <-> wb 176  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859  (/)c0 3550  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-sn 3741

This theorem is referenced by:  snex  4111  prprc2  4122  0nel1c  4159  snfi  4431  imasn  5018  dmsnopss  5067  fconst5  5455  ecexr  5950  frecxp  6314