Theorem snprc 1838

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 ∈ V ↔ {A} = ∅)

Proof of Theorem snprc

Step	Hyp	Ref	Expression
1		elsn 1820	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
2	1	biex 733	. . 3 ⊢ (∃x x ∈ {A} ↔ ∃x x = A)
3		n0 1714	. . 3 ⊢ (¬ {A} = ∅ ↔ ∃x x ∈ {A})
4		isset 1351	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
5	2, 3, 4	3bitr4 158	. 2 ⊢ (¬ {A} = ∅ ↔ A ∈ V)
6	5	bicon1i 193	1 ⊢ (¬ A ∈ V ↔ {A} = ∅)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808

This theorem is referenced by:  prprc 1839  snsspr 1853  snex 1859  opprc1 1905  opprc3 1908  sucprc 2297  dmsnop 2547  imasn 2616  fvprc 2829  1stval 3089  2ndval 3090  snfi 3337

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-nul 1708  df-sn 1811

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