Theorem inindif

Description: See inundif . (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	1	orci	⊢ ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶 )
3		inss	⊢ ( ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 ∨ 𝐴 ⊆ 𝐶 ) → ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶 )
4	2 3	ax-mp	⊢ ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶
5		inssdif0	⊢ ( ( ( 𝐴 ∩ 𝐶 ) ∩ 𝐴 ) ⊆ 𝐶 ↔ ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐴 ∖ 𝐶 ) ) = ∅ )
6	4 5	mpbi	⊢ ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐴 ∖ 𝐶 ) ) = ∅