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Theorem snprc 3462
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 3419 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1512 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
32notbii 604 . 2 (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4 eq0 3266 . . 3 ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5 alnex 1404 . . 3 (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
64, 5bitri 177 . 2 ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7 isset 2578 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
87notbii 604 . 2 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
93, 6, 83bitr4ri 206 1 𝐴 ∈ V ↔ {𝐴} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  c0 3251  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-nul 3252  df-sn 3408
This theorem is referenced by:  prprc1  3505  prprc  3507  snexprc  3965  sucprc  4176  snnen2oprc  6353
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