Theorem nvelim 42726

Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5077. (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 5077	. 2 ⊢ ¬ V ∈ 𝐵
2		eleq1 2854	. . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	eqcoms 2787	. 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	1, 3	mtbii 318	1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1507   ∈ wcel 2050  Vcvv 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2751  ax-sep 5060

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-v 3418

This theorem is referenced by:  afvvdm  42744  afvvfunressn  42746  afvvv  42748  afvvfveq  42751