Theorem bj-disj2r 37011

Description: Relative version of ssdifin0 4451, allowing a biconditional, and of disj2 4423. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4451 nor disj2 4423. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-disj2r	⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)

Proof of Theorem bj-disj2r

Step	Hyp	Ref	Expression
1		dfss2 3934	. . 3 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉))
2		indif2 4246	. . . . 5 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵)
3		inss1 4202	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
4		ssid 3971	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
5		inss2 4203	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ 𝑉
6	4, 5	ssini 4205	. . . . . . 7 ⊢ (𝐴 ∩ 𝑉) ⊆ ((𝐴 ∩ 𝑉) ∩ 𝑉)
7	3, 6	eqssi 3965	. . . . . 6 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) = (𝐴 ∩ 𝑉)
8	7	difeq1i 4087	. . . . 5 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
9	2, 8	eqtri 2753	. . . 4 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
10	9	eqeq1i 2735	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉) ↔ ((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉))
11		eqcom 2737	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
12	1, 10, 11	3bitri 297	. 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
13		disj3 4419	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
14		in32 4195	. . 3 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝑉)
15	14	eqeq1i 2735	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
16	12, 13, 15	3bitr2i 299	1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1540   ∖ cdif 3913   ∩ cin 3915   ⊆ wss 3916  ∅c0 4298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3919  df-in 3923  df-ss 3933  df-nul 4299

This theorem is referenced by:  bj-sscon  37012