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Metamath Proof Explorer

Theorem opelxpii

Description: Ordered pair membership in a Cartesian product (implication), induction form. (Contributed by Steven Nguyen, 17-Jul-2022)

Ref Expression
Hypotheses opelxpii.1 ⊢ 𝐴 ∈ 𝐶
opelxpii.2 ⊢ 𝐵 ∈ 𝐷
Assertion opelxpii ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 )

Proof

Step Hyp Ref Expression
1 opelxpii.1 ⊢ 𝐴 ∈ 𝐶
2 opelxpii.2 ⊢ 𝐵 ∈ 𝐷
3 opelxpi ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )
4 1 2 3 mp2an ⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 )