Theorem bj-disj2r 37513

Description: Relative version of ssdifin0 4439, allowing a biconditional, and of disj2 4412. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4439 nor disj2 4412. (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 3922	. . 3 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉))
2		indif2 4233	. . . . 5 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵)
3		inss1 4188	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
4		ssid 3958	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
5		inss2 4189	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ 𝑉
6	4, 5	ssini 4191	. . . . . . 7 ⊢ (𝐴 ∩ 𝑉) ⊆ ((𝐴 ∩ 𝑉) ∩ 𝑉)
7	3, 6	eqssi 3952	. . . . . 6 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) = (𝐴 ∩ 𝑉)
8	7	difeq1i 4076	. . . . 5 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
9	2, 8	eqtri 2785	. . . 4 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
10	9	eqeq1i 2767	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉) ↔ ((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉))
11		eqcom 2769	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
12	1, 10, 11	3bitri 299	. 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
13		disj3 4408	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
14		in32 4181	. . 3 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝑉)
15	14	eqeq1i 2767	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
16	12, 13, 15	3bitr2i 301	1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)

Syntax hints:   ↔ wb 208   = wceq 1560   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286

This theorem is referenced by:  bj-sscon  37514