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Theorem nev 37582
 Description: Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
nev (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nev
StepHypRef Expression
1 eqv 3195 . 2 (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
21necon3abii 2836 1 (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1478   ∈ wcel 1987   ≠ wne 2790  Vcvv 3190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3192 This theorem is referenced by: (None)
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