Theorem difininv 32010

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

Assertion

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

Proof of Theorem difininv

Step	Hyp	Ref	Expression
1		indif1 4271	. . . . . 6 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
2	1	eqeq1i 2737	. . . . 5 ⊢ (((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ∅)
3		ssdif0 4363	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ∅)
4	2, 3	sylbb2 237	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5	4	adantr 481	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ 𝐶)
6		inss2 4229	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	5, 7	ssind 4232	. 2 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐵))
9		indif1 4271	. . . . . 6 ⊢ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∖ 𝐴)
10	9	eqeq1i 2737	. . . . 5 ⊢ (((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅ ↔ ((𝐶 ∩ 𝐵) ∖ 𝐴) = ∅)
11		ssdif0 4363	. . . . 5 ⊢ ((𝐶 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐶 ∩ 𝐵) ∖ 𝐴) = ∅)
12	10, 11	sylbb2 237	. . . 4 ⊢ (((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅ → (𝐶 ∩ 𝐵) ⊆ 𝐴)
13	12	adantl 482	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ 𝐴)
14		inss2 4229	. . . 4 ⊢ (𝐶 ∩ 𝐵) ⊆ 𝐵
15	14	a1i 11	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ 𝐵)
16	13, 15	ssind 4232	. 2 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ (𝐴 ∩ 𝐵))
17	8, 16	eqssd 3999	1 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  chtvalz  33927