Theorem disjdifr

Description: A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023)

	Ref	Expression
Assertion	disjdifr	⊢ ( ( 𝐵 ∖ 𝐴 ) ∩ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		incom	⊢ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ( ( 𝐵 ∖ 𝐴 ) ∩ 𝐴 )
2		disjdif	⊢ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅
3	1 2	eqtr3i	⊢ ( ( 𝐵 ∖ 𝐴 ) ∩ 𝐴 ) = ∅