Theorem ndmov

Description: The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995)

	Ref	Expression
Hypothesis	ndmov.1	⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
Assertion	ndmov	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		ndmov.1	⊢ dom 𝐹 = ( 𝑆 × 𝑆 )
2		ndmovg	⊢ ( ( dom 𝐹 = ( 𝑆 × 𝑆 ) ∧ ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ∅ )
3	1 2	mpan	⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) = ∅ )