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Metamath Proof Explorer

Theorem nel2nelin

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

Ref Expression
Assertion nel2nelin ( ¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 elinel2 ⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐶 )
2 1 con3i ⊢ ( ¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )