Theorem nvelim 44867

Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5255. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
nvelim	⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Proof of Theorem nvelim

Step	Hyp	Ref	Expression
1		nvel 5255	. 2 ⊢ ¬ V ∈ 𝐵
2		eleq1 2825	. . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	eqcoms 2745	. 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	1, 3	mtbii 325	1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-sep 5238

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443

This theorem is referenced by:  afvvdm  44885  afvvfunressn  44887  afvvv  44889  afvvfveq  44892