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Theorem snprc 3558
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 3514 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1569 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
32notbii 642 . 2 (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4 eq0 3351 . . 3 ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5 alnex 1460 . . 3 (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
64, 5bitri 183 . 2 ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7 isset 2666 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
87notbii 642 . 2 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
93, 6, 83bitr4ri 212 1 𝐴 ∈ V ↔ {𝐴} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1314   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  c0 3333  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334  df-sn 3503
This theorem is referenced by:  prprc1  3601  prprc  3603  snexprc  4080  sucprc  4304  snnen2oprc  6722  unsnfidcex  6776
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