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Theorem bj-disj2r 35528
Description: Relative version of ssdifin0 4448, allowing a biconditional, and of disj2 4422. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4448 nor disj2 4422. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-disj2r ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Proof of Theorem bj-disj2r
StepHypRef Expression
1 df-ss 3932 . . 3 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉))
2 indif2 4235 . . . . 5 ((𝐴𝑉) ∩ (𝑉𝐵)) = (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵)
3 inss1 4193 . . . . . . 7 ((𝐴𝑉) ∩ 𝑉) ⊆ (𝐴𝑉)
4 ssid 3971 . . . . . . . 8 (𝐴𝑉) ⊆ (𝐴𝑉)
5 inss2 4194 . . . . . . . 8 (𝐴𝑉) ⊆ 𝑉
64, 5ssini 4196 . . . . . . 7 (𝐴𝑉) ⊆ ((𝐴𝑉) ∩ 𝑉)
73, 6eqssi 3965 . . . . . 6 ((𝐴𝑉) ∩ 𝑉) = (𝐴𝑉)
87difeq1i 4083 . . . . 5 (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴𝑉) ∖ 𝐵)
92, 8eqtri 2765 . . . 4 ((𝐴𝑉) ∩ (𝑉𝐵)) = ((𝐴𝑉) ∖ 𝐵)
109eqeq1i 2742 . . 3 (((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉) ↔ ((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉))
11 eqcom 2744 . . 3 (((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
121, 10, 113bitri 297 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
13 disj3 4418 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
14 in32 4186 . . 3 ((𝐴𝑉) ∩ 𝐵) = ((𝐴𝐵) ∩ 𝑉)
1514eqeq1i 2742 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
1612, 13, 153bitr2i 299 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  cdif 3912  cin 3914  wss 3915  c0 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-in 3922  df-ss 3932  df-nul 4288
This theorem is referenced by:  bj-sscon  35529
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