Theorem snprc 2439

Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48.

Assertion

Ref	Expression
snprc	|- (-. A e. V <-> {A} = (/))

Proof of Theorem snprc

Step	Hyp	Ref	Expression
1		elsn 2417	. . . 4 |- (x e. {A} <-> x = A)
2	1	exbii 1049	. . 3 |- (E.x x e. {A} <-> E.x x = A)
3		n0 2285	. . 3 |- (-. {A} = (/) <-> E.x x e. {A})
4		isset 1810	. . 3 |- (A e. V <-> E.x x = A)
5	2, 3, 4	3bitr4 183	. 2 |- (-. {A} = (/) <-> A e. V)
6	5	con1bii 220	1 |- (-. A e. V <-> {A} = (/))

Syntax hints:  -. wn 2   <-> wb 146   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807  (/)c0 2276  {csn 2405

This theorem is referenced by:  prprc1 2448  prprc 2450  snsspr 2466  opprc1 2494  snex 2745  opprc3 2792  unisn2 2870  sucprc 3039  dmsnop 3323  relimasn 3417  fvprc 3712  fconst5 3839  1stval 4071  2ndval 4072  snfi 4419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-nul 2277  df-sn 2408

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