Theorem disjr

Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011)

	Ref	Expression
Assertion	disjr	$$⊢A\cap B=\varnothing \leftrightarrow \forall x\in B\neg x\in A$$

Proof

Step	Hyp	Ref	Expression
1		incom	$$⊢A\cap B=B\cap A$$
2	1	eqeq1i	$$⊢A\cap B=\varnothing \leftrightarrow B\cap A=\varnothing$$
3		disj	$$⊢B\cap A=\varnothing \leftrightarrow \forall x\in B\neg x\in A$$
4	2 3	bitri	$$⊢A\cap B=\varnothing \leftrightarrow \forall x\in B\neg x\in A$$