Theorem nev 42825

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 3481	. 2 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
2	1	necon3abii 2985	1 ⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-v 3474

This theorem is referenced by: (None)