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Theorem bj-disj2r 34356
Description: Relative version of ssdifin0 4404, allowing a biconditional, and of disj2 4380. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssdifin0 4404 nor disj2 4380. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-disj2r ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)

Proof of Theorem bj-disj2r
StepHypRef Expression
1 df-ss 3927 . . 3 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉))
2 indif2 4222 . . . . 5 ((𝐴𝑉) ∩ (𝑉𝐵)) = (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵)
3 inss1 4180 . . . . . . 7 ((𝐴𝑉) ∩ 𝑉) ⊆ (𝐴𝑉)
4 ssid 3965 . . . . . . . 8 (𝐴𝑉) ⊆ (𝐴𝑉)
5 inss2 4181 . . . . . . . 8 (𝐴𝑉) ⊆ 𝑉
64, 5ssini 4183 . . . . . . 7 (𝐴𝑉) ⊆ ((𝐴𝑉) ∩ 𝑉)
73, 6eqssi 3959 . . . . . 6 ((𝐴𝑉) ∩ 𝑉) = (𝐴𝑉)
87difeq1i 4071 . . . . 5 (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴𝑉) ∖ 𝐵)
92, 8eqtri 2844 . . . 4 ((𝐴𝑉) ∩ (𝑉𝐵)) = ((𝐴𝑉) ∖ 𝐵)
109eqeq1i 2826 . . 3 (((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉) ↔ ((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉))
11 eqcom 2828 . . 3 (((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
121, 10, 113bitri 300 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
13 disj3 4376 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
14 in32 4173 . . 3 ((𝐴𝑉) ∩ 𝐵) = ((𝐴𝐵) ∩ 𝑉)
1514eqeq1i 2826 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
1612, 13, 153bitr2i 302 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  cdif 3907  cin 3909  wss 3910  c0 4266
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4267
This theorem is referenced by:  bj-sscon  34357
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