Theorem vneqv 5251

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5252 and shorten proof. (Revised by BJ, 25-Apr-2026.)

Assertion

Ref	Expression
vneqv	⊢ ¬ 𝑥 = V

Proof of Theorem vneqv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3433	. . . 4 ⊢ 𝑦 ∈ V
2		eleq2 2825	. . . 4 ⊢ (𝑥 = V → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 258	. . 3 ⊢ (𝑥 = V → 𝑦 ∈ 𝑥)
4	3	con3i 154	. 2 ⊢ (¬ 𝑦 ∈ 𝑥 → ¬ 𝑥 = V)
5		exnelv 5248	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
6	4, 5	exlimiiv 1933	1 ⊢ ¬ 𝑥 = V

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431

This theorem is referenced by:  vnex  5252