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Theorem notnotsnex 4184
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 4181 . . . . 5 (𝐴 ∈ V → {𝐴} ∈ V)
21con3i 632 . . . 4 (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3 snexprc 4183 . . . 4 𝐴 ∈ V → {𝐴} ∈ V)
42, 3syl 14 . . 3 (¬ {𝐴} ∈ V → {𝐴} ∈ V)
54con3i 632 . 2 (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6 pm2.01 616 . 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 2148  Vcvv 2737  {csn 3591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597
This theorem is referenced by: (None)
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