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Theorem notnotsnex 4202
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 4199 . . . . 5 (𝐴 ∈ V → {𝐴} ∈ V)
21con3i 633 . . . 4 (¬ {𝐴} ∈ V → ¬ 𝐴 ∈ V)
3 snexprc 4201 . . . 4 𝐴 ∈ V → {𝐴} ∈ V)
42, 3syl 14 . . 3 (¬ {𝐴} ∈ V → {𝐴} ∈ V)
54con3i 633 . 2 (¬ {𝐴} ∈ V → ¬ ¬ {𝐴} ∈ V)
6 pm2.01 617 . 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 2160  Vcvv 2752  {csn 3607
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613
This theorem is referenced by: (None)
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