Theorem nel2nelini 45332

Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
nel2nelini.1	⊢ ¬ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
nel2nelini	⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

Proof of Theorem nel2nelini

Step	Hyp	Ref	Expression
1		nel2nelini.1	. 2 ⊢ ¬ 𝐴 ∈ 𝐶
2		nel2nelin 4158	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:  ¬ wn 3   ∈ wcel 2113   ∩ cin 3898

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 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-in 3906

This theorem is referenced by: (None)