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Theorem bj-disj2r 33585
Description: Relative version of ssdifin0 4273, allowing a biconditional, and of disj2 4249. This proof does not rely, even indirectly, on ssdifin0 4273 nor disj2 4249. (Contributed by BJ, 11-Nov-2021.)
Assertion
Ref Expression
bj-disj2r ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Proof of Theorem bj-disj2r
StepHypRef Expression
1 df-ss 3805 . . 3 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉))
2 indif2 4096 . . . . 5 ((𝐴𝑉) ∩ (𝑉𝐵)) = (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵)
3 inss1 4052 . . . . . . 7 ((𝐴𝑉) ∩ 𝑉) ⊆ (𝐴𝑉)
4 ssid 3841 . . . . . . . 8 (𝐴𝑉) ⊆ (𝐴𝑉)
5 inss2 4053 . . . . . . . 8 (𝐴𝑉) ⊆ 𝑉
64, 5ssini 4055 . . . . . . 7 (𝐴𝑉) ⊆ ((𝐴𝑉) ∩ 𝑉)
73, 6eqssi 3836 . . . . . 6 ((𝐴𝑉) ∩ 𝑉) = (𝐴𝑉)
87difeq1i 3946 . . . . 5 (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴𝑉) ∖ 𝐵)
92, 8eqtri 2801 . . . 4 ((𝐴𝑉) ∩ (𝑉𝐵)) = ((𝐴𝑉) ∖ 𝐵)
109eqeq1i 2782 . . 3 (((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉) ↔ ((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉))
11 eqcom 2784 . . 3 (((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
121, 10, 113bitri 289 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
13 disj3 4245 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
14 in32 4045 . . 3 ((𝐴𝑉) ∩ 𝐵) = ((𝐴𝐵) ∩ 𝑉)
1514eqeq1i 2782 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
1612, 13, 153bitr2i 291 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  cdif 3788  cin 3790  wss 3791  c0 4140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rab 3098  df-v 3399  df-dif 3794  df-in 3798  df-ss 3805  df-nul 4141
This theorem is referenced by:  bj-sscon  33586
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