Theorem vneqv 5260

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5261 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 3452	. . . 4 ⊢ 𝑦 ∈ V
2		eleq2 2845	. . . 4 ⊢ (𝑥 = V → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
3	1, 2	mpbiri 260	. . 3 ⊢ (𝑥 = V → 𝑦 ∈ 𝑥)
4	3	con3i 154	. 2 ⊢ (¬ 𝑦 ∈ 𝑥 → ¬ 𝑥 = V)
5		exnelv 5257	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
6	4, 5	exlimiiv 1945	1 ⊢ ¬ 𝑥 = V

Syntax hints:  ¬ wn 3   = wceq 1554   ∈ wcel 2136  Vcvv 3448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450

This theorem is referenced by:  vnex  5261