Theorem nvelim 47154

Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5249. (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 5249	. 2 ⊢ ¬ V ∈ 𝐵
2		eleq1 2819	. . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	eqcoms 2739	. 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	1, 3	mtbii 326	1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438

This theorem is referenced by:  afvvdm  47172  afvvfunressn  47174  afvvv  47176  afvvfveq  47179