Theorem disjr

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

	Ref	Expression
Assertion	disjr	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		incom	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
2	1	eqeq1i	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ( 𝐵 ∩ 𝐴 ) = ∅ )
3		disj	⊢ ( ( 𝐵 ∩ 𝐴 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )
4	2 3	bitri	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )