Theorem bj-disj2r 37941

Description: Relative version of ssdifin0 4440, allowing a biconditional, and of disj2 4410. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4440 nor disj2 4410. (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 4094	. . 3 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉))
2		indif2 4226	. . . . 5 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵)
3		inss1 4133	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
4		ssid 3930	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ (𝐴 ∩ 𝑉)
5		inss2 4134	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑉) ⊆ 𝑉
6	4, 5	ssini 4136	. . . . . . 7 ⊢ (𝐴 ∩ 𝑉) ⊆ ((𝐴 ∩ 𝑉) ∩ 𝑉)
7	3, 6	eqssi 3924	. . . . . 6 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝑉) = (𝐴 ∩ 𝑉)
8	7	difeq1i 4059	. . . . 5 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
9	2, 8	eqtri 2783	. . . 4 ⊢ ((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = ((𝐴 ∩ 𝑉) ∖ 𝐵)
10	9	eqeq1i 2765	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∩ (𝑉 ∖ 𝐵)) = (𝐴 ∩ 𝑉) ↔ ((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉))
11		eqcom 2767	. . 3 ⊢ (((𝐴 ∩ 𝑉) ∖ 𝐵) = (𝐴 ∩ 𝑉) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
12	1, 10, 11	3bitri 300	. 2 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
13		disj3 4406	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ (𝐴 ∩ 𝑉) = ((𝐴 ∩ 𝑉) ∖ 𝐵))
14		in32 4126	. . 3 ⊢ ((𝐴 ∩ 𝑉) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝑉)
15	14	eqeq1i 2765	. 2 ⊢ (((𝐴 ∩ 𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)
16	12, 13, 15	3bitr2i 302	1 ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279

This theorem is used by:  bj-sscon  37942