Theorem ndmovcom

Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995)

	Ref	Expression
Hypothesis	ndmov.1	$$⊢\mathrm{dom}F=S\times S$$
Assertion	ndmovcom	$$⊢\neg \left(A\in S\wedge B\in S\right)\to AFB=BFA$$

Proof

Step	Hyp	Ref	Expression
1		ndmov.1	$$⊢\mathrm{dom}F=S\times S$$
2	1	ndmov	$$⊢\neg \left(A\in S\wedge B\in S\right)\to AFB=\varnothing$$
3		ancom	$$⊢\left(A\in S\wedge B\in S\right)\leftrightarrow \left(B\in S\wedge A\in S\right)$$
4	1	ndmov	$$⊢\neg \left(B\in S\wedge A\in S\right)\to BFA=\varnothing$$
5	3 4	sylnbi	$$⊢\neg \left(A\in S\wedge B\in S\right)\to BFA=\varnothing$$
6	2 5	eqtr4d	$$⊢\neg \left(A\in S\wedge B\in S\right)\to AFB=BFA$$