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Theorem snprc 3502
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 3458 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1541 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
32notbii 629 . 2 (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4 eq0 3299 . . 3 ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5 alnex 1433 . . 3 (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
64, 5bitri 182 . 2 ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7 isset 2625 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
87notbii 629 . 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 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  c0 3284  {csn 3441
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-sn 3447
This theorem is referenced by:  prprc1  3545  prprc  3547  snexprc  4012  sucprc  4230  snnen2oprc  6556  unsnfidcex  6610
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