Theorem nev 43759

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 3487	. 2 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
2	1	necon3abii 2984	1 ⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1534   ∈ wcel 2105   ≠ wne 2937  Vcvv 3477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479

This theorem is referenced by: (None)