Theorem difininv

Description: Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021)

	Ref	Expression
Assertion	difininv	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) = ( 𝐶 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		indif1	⊢ ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )
2	1	eqeq1i	⊢ ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ↔ ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 ) = ∅ )
3		ssdif0	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ↔ ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 ) = ∅ )
4	2 3	sylbb2	⊢ ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )
5	4	adantr	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )
6		inss2	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵
7	6	a1i	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵 )
8	5 7	ssind	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) ⊆ ( 𝐶 ∩ 𝐵 ) )
9		indif1	⊢ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ( ( 𝐶 ∩ 𝐵 ) ∖ 𝐴 )
10	9	eqeq1i	⊢ ( ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ↔ ( ( 𝐶 ∩ 𝐵 ) ∖ 𝐴 ) = ∅ )
11		ssdif0	⊢ ( ( 𝐶 ∩ 𝐵 ) ⊆ 𝐴 ↔ ( ( 𝐶 ∩ 𝐵 ) ∖ 𝐴 ) = ∅ )
12	10 11	sylbb2	⊢ ( ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ → ( 𝐶 ∩ 𝐵 ) ⊆ 𝐴 )
13	12	adantl	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐶 ∩ 𝐵 ) ⊆ 𝐴 )
14		inss2	⊢ ( 𝐶 ∩ 𝐵 ) ⊆ 𝐵
15	14	a1i	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐶 ∩ 𝐵 ) ⊆ 𝐵 )
16	13 15	ssind	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐶 ∩ 𝐵 ) ⊆ ( 𝐴 ∩ 𝐵 ) )
17	8 16	eqssd	⊢ ( ( ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ∅ ∧ ( ( 𝐶 ∖ 𝐴 ) ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) = ( 𝐶 ∩ 𝐵 ) )