Theorem vneqv 5278

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5279 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 3458	. . . 4 ⊢ 𝑦 ∈ V
2		eleq2 2851	. . . 4 ⊢ (𝑥 = V → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 261	. . 3 ⊢ (𝑥 = V → 𝑦 ∈ 𝑥)
4	3	con3i 155	. 2 ⊢ (¬ 𝑦 ∈ 𝑥 → ¬ 𝑥 = V)
5		exnelv 5275	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
6	4, 5	exlimiiv 1960	1 ⊢ ¬ 𝑥 = V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456

This theorem is used by:  vnex  5279