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Theorem snprc 3556
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 3512 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1567 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
32notbii 640 . 2 (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4 eq0 3349 . . 3 ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5 alnex 1458 . . 3 (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
64, 5bitri 183 . 2 ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7 isset 2664 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
87notbii 640 . 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 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  c0 3331  {csn 3495
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-nul 3332  df-sn 3501
This theorem is referenced by:  prprc1  3599  prprc  3601  snexprc  4078  sucprc  4302  snnen2oprc  6720  unsnfidcex  6774
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