Theorem nvelim 41200

Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4797. (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 4797	. 2 ⊢ ¬ V ∈ 𝐵
2		eleq1 2689	. . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	eqcoms 2630	. 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	1, 3	mtbii 316	1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  afvvdm  41221  afvvfunressn  41223  afvvv  41225  afvvfveq  41228