Theorem vneqv 5273

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5274 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 2853	. . . 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 5270	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
6	4, 5	exlimiiv 1966	1 ⊢ ¬ 𝑥 = V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  vnex  5274