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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 V ↔ {A} = )

Proof of Theorem snprc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elsn 3748 . . . 4 (x {A} ↔ x = A)
21exbii 1582 . . 3 (x x {A} ↔ x x = A)
3 neq0 3560 . . 3 (¬ {A} = x x {A})
4 isset 2863 . . 3 (A V ↔ x x = A)
52, 3, 43bitr4i 268 . 2 (¬ {A} = A V)
65con1bii 321 1 A V ↔ {A} = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∅c0 3550  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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