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Theorem snprc 3484
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 velsn 3442 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1539 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
32notbii 627 . 2 (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4 eq0 3287 . . 3 ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5 alnex 1431 . . 3 (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
64, 5bitri 182 . 2 ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7 isset 2618 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
87notbii 627 . 2 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
93, 6, 83bitr4ri 211 1 𝐴 ∈ V ↔ {𝐴} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wal 1285   = wceq 1287  wex 1424  wcel 1436  Vcvv 2614  c0 3272  {csn 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-nul 3273  df-sn 3431
This theorem is referenced by:  prprc1  3527  prprc  3529  snexprc  3988  sucprc  4206  snnen2oprc  6509  unsnfidcex  6560
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