This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 } ) ) = ∅