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Theorem notnotsnex 3989
Description: A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
Assertion
Ref Expression
notnotsnex ¬ ¬ {𝐴} ∈ V

Proof of Theorem notnotsnex
StepHypRef Expression
1 snexg 3986 . . . . 5 (𝐴 ∈ V → {𝐴} ∈ V)
21con3i 595 . . . 4 (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3 snexprc 3988 . . . 4 𝐴 ∈ V → {𝐴} ∈ V)
42, 3syl 14 . . 3 (¬ {𝐴} ∈ V → {𝐴} ∈ V)
54con3i 595 . 2 (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6 ax-in1 577 . 2 ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
75, 6ax-mp 7 1 ¬ ¬ {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1436  Vcvv 2614  {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-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977
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-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431
This theorem is referenced by: (None)
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