Theorem xp01disj

Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007)

	Ref	Expression
Assertion	xp01disj	⊢ ( ( 𝐴 × { ∅ } ) ∩ ( 𝐶 × { 1o } ) ) = ∅

Proof

Step	Hyp	Ref	Expression
1		1n0	⊢ 1o ≠ ∅
2	1	necomi	⊢ ∅ ≠ 1o
3		xpsndisj	⊢ ( ∅ ≠ 1o → ( ( 𝐴 × { ∅ } ) ∩ ( 𝐶 × { 1o } ) ) = ∅ )
4	2 3	ax-mp	⊢ ( ( 𝐴 × { ∅ } ) ∩ ( 𝐶 × { 1o } ) ) = ∅