Theorem nel2nelini 43824

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 43822	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:  ¬ wn 3   ∈ wcel 2107   ∩ cin 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3955

This theorem is referenced by: (None)