Theorem nvelim 47598

Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5243. (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 5243	. 2 ⊢ ¬ V ∈ 𝐵
2		eleq1 2829	. . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	eqcoms 2749	. 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	1, 3	mtbii 328	1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  Vcvv 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435

This theorem is referenced by:  afvvdm  47616  afvvfunressn  47618  afvvv  47620  afvvfveq  47623