Theorem snprc 3789

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 ∈ V ↔ {A} = ∅)

Proof of Theorem snprc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsn 3749	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
2	1	exbii 1582	. . 3 ⊢ (∃x x ∈ {A} ↔ ∃x x = A)
3		neq0 3561	. . 3 ⊢ (¬ {A} = ∅ ↔ ∃x x ∈ {A})
4		isset 2864	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
5	2, 3, 4	3bitr4i 268	. 2 ⊢ (¬ {A} = ∅ ↔ A ∈ V)
6	5	con1bii 321	1 ⊢ (¬ A ∈ V ↔ {A} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ∅c0 3551  {csn 3738

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-sn 3742

This theorem is referenced by:  snex  4112  prprc2  4123  0nel1c  4160  snfi  4432  imasn  5019  dmsnopss  5068  fconst5  5456  ecexr  5951  frecxp  6315