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Theorem notnotsnex 4121
 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 4118 . . . . 5 (𝐴 ∈ V → {𝐴} ∈ V)
21con3i 622 . . . 4 (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3 snexprc 4120 . . . 4 𝐴 ∈ V → {𝐴} ∈ V)
42, 3syl 14 . . 3 (¬ {𝐴} ∈ V → {𝐴} ∈ V)
54con3i 622 . 2 (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6 pm2.01 606 . 2 ((¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V) → ¬ ¬ {𝐴} ∈ V)
75, 6ax-mp 5 1 ¬ ¬ {𝐴} ∈ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2112  Vcvv 2691  {csn 3534 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-nul 4064  ax-pow 4108 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-dif 3080  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540 This theorem is referenced by: (None)
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