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Metamath Proof Explorer

Theorem elini

Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses elini.1 ⊢ 𝐴 ∈ 𝐵
elini.2 ⊢ 𝐴 ∈ 𝐶
Assertion elini 𝐴 ∈ ( 𝐵 ∩ 𝐶 )

Proof

Step Hyp Ref Expression
1 elini.1 ⊢ 𝐴 ∈ 𝐵
2 elini.2 ⊢ 𝐴 ∈ 𝐶
3 elin ⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
4 1 2 3 mpbir2an ⊢ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )