Theorem nev 39011

Description: Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)

Assertion

Ref	Expression
nev	⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem nev

Step	Hyp	Ref	Expression
1		eqv 3404	. 2 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
2	1	necon3abii 3014	1 ⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1599   ∈ wcel 2106   ≠ wne 2968  Vcvv 3397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-12 2162  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-ne 2969  df-v 3399

This theorem is referenced by: (None)