Theorem inindif

Description: The intersection and class difference of a class with another class are disjoint. With inundif , this shows that such intersection and class difference partition the class A . (Contributed by Thierry Arnoux, 13-Sep-2017)

	Ref	Expression
Assertion	inindif	⊢ ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐴 ∖ 𝐶 ) ) = ∅

Proof

Step	Hyp	Ref	Expression
1		inss2	⊢ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶
2		ssinss1	⊢ ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 → ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶 )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶
4		inssdif0	⊢ ( ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶 ↔ ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐴 ∖ 𝐶 ) ) = ∅ )
5	3 4	mpbi	⊢ ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐴 ∖ 𝐶 ) ) = ∅