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Theorem bj-disj2r 35897
Description: Relative version of ssdifin0 4484, allowing a biconditional, and of disj2 4456. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4484 nor disj2 4456. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-disj2r ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Proof of Theorem bj-disj2r
StepHypRef Expression
1 df-ss 3964 . . 3 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉))
2 indif2 4269 . . . . 5 ((𝐴𝑉) ∩ (𝑉𝐵)) = (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵)
3 inss1 4227 . . . . . . 7 ((𝐴𝑉) ∩ 𝑉) ⊆ (𝐴𝑉)
4 ssid 4003 . . . . . . . 8 (𝐴𝑉) ⊆ (𝐴𝑉)
5 inss2 4228 . . . . . . . 8 (𝐴𝑉) ⊆ 𝑉
64, 5ssini 4230 . . . . . . 7 (𝐴𝑉) ⊆ ((𝐴𝑉) ∩ 𝑉)
73, 6eqssi 3997 . . . . . 6 ((𝐴𝑉) ∩ 𝑉) = (𝐴𝑉)
87difeq1i 4117 . . . . 5 (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴𝑉) ∖ 𝐵)
92, 8eqtri 2760 . . . 4 ((𝐴𝑉) ∩ (𝑉𝐵)) = ((𝐴𝑉) ∖ 𝐵)
109eqeq1i 2737 . . 3 (((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉) ↔ ((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉))
11 eqcom 2739 . . 3 (((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
121, 10, 113bitri 296 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
13 disj3 4452 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
14 in32 4220 . . 3 ((𝐴𝑉) ∩ 𝐵) = ((𝐴𝐵) ∩ 𝑉)
1514eqeq1i 2737 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
1612, 13, 153bitr2i 298 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  cdif 3944  cin 3946  wss 3947  c0 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322
This theorem is referenced by:  bj-sscon  35898
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