Theorem difininv 29872

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		ssdif0 4142	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ∅)
2		indif1 4072	. . . . . . 7 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3	2	eqeq1i 2804	. . . . . 6 ⊢ (((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ∅)
4	1, 3	bitr4i 270	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅)
5	4	biimpri 220	. . . 4 ⊢ (((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ 𝐶)
6	5	adantr 473	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ 𝐶)
7		inss2 4029	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
8	7	a1i 11	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ 𝐵)
9	6, 8	ssind 4032	. 2 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐵))
10		ssdif0 4142	. . . . . 6 ⊢ ((𝐶 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐶 ∩ 𝐵) ∖ 𝐴) = ∅)
11		indif1 4072	. . . . . . 7 ⊢ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∖ 𝐴)
12	11	eqeq1i 2804	. . . . . 6 ⊢ (((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅ ↔ ((𝐶 ∩ 𝐵) ∖ 𝐴) = ∅)
13	10, 12	bitr4i 270	. . . . 5 ⊢ ((𝐶 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅)
14	13	biimpri 220	. . . 4 ⊢ (((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅ → (𝐶 ∩ 𝐵) ⊆ 𝐴)
15	14	adantl 474	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ 𝐴)
16		inss2 4029	. . . 4 ⊢ (𝐶 ∩ 𝐵) ⊆ 𝐵
17	16	a1i 11	. . 3 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ 𝐵)
18	15, 17	ssind 4032	. 2 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐵) ⊆ (𝐴 ∩ 𝐵))
19	9, 18	eqssd 3815	1 ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∖ cdif 3766   ∩ cin 3768   ⊆ wss 3769  ∅c0 4115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116

This theorem is referenced by:  chtvalz  31227