Theorem nev 44214

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 3441	. 2 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
2	1	necon3abii 2980	1 ⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 207  ∀wal 1545   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433

This theorem is referenced by: (None)