Theorem nel2nelini 45905

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 4164	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146   ∩ cin 3907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915

This theorem is used by: (None)