Theorem xpdisj1

Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004)

	Ref	Expression
Assertion	xpdisj1	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐴 × 𝐶 ) ∩ ( 𝐵 × 𝐷 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		xpeq1	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐴 ∩ 𝐵 ) × ( 𝐶 ∩ 𝐷 ) ) = ( ∅ × ( 𝐶 ∩ 𝐷 ) ) )
2		inxp	⊢ ( ( 𝐴 × 𝐶 ) ∩ ( 𝐵 × 𝐷 ) ) = ( ( 𝐴 ∩ 𝐵 ) × ( 𝐶 ∩ 𝐷 ) )
3		0xp	⊢ ( ∅ × ( 𝐶 ∩ 𝐷 ) ) = ∅
4	3	eqcomi	⊢ ∅ = ( ∅ × ( 𝐶 ∩ 𝐷 ) )
5	1 2 4	3eqtr4g	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐴 × 𝐶 ) ∩ ( 𝐵 × 𝐷 ) ) = ∅ )