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		ineqcom	$$⊢A\cap B=\varnothing \leftrightarrow B\cap A=\varnothing$$
2		disj	$$⊢B\cap A=\varnothing \leftrightarrow \forall x\in B\neg x\in A$$
3	1 2	bitri	$$⊢A\cap B=\varnothing \leftrightarrow \forall x\in B\neg x\in A$$