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Metamath Proof Explorer

Theorem opelxpd

Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021)

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

Proof

Step Hyp Ref Expression
1 opelxpd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2 opelxpd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
3 opelxpi ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 × 𝐷 ) )