Theorem vneqv 5271

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5272 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 3461	. . . 4 ⊢ 𝑦 ∈ V
2		eleq2 2854	. . . 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 5268	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
6	4, 5	exlimiiv 1954	1 ⊢ ¬ 𝑥 = V

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is referenced by:  vnex  5272