Theorem elxp

Description: Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994)

	Ref	Expression
Assertion	elxp	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-xp	⊢ ( 𝐵 × 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) }
2	1	eleq2i	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) ↔ 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) } )
3		elopab	⊢ ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
4	2 3	bitri	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )