Metamath Proof Explorer


Theorem xpdisj1

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

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

Proof

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