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Metamath Proof Explorer

Theorem elinel2

Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion elinel2 ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐶 )

Proof

Step Hyp Ref Expression
1 elin ⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
2 1 simprbi ⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐶 )