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Theorem bj-disj2r 33338
 Description: Relative version of ssdifin0 4190, allowing a biconditional, and of disj2 4166. This proof does not rely, even indirectly, on ssdifin0 4190 nor disj2 4166. (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 3735 . . 3 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉))
2 indif2 4017 . . . . 5 ((𝐴𝑉) ∩ (𝑉𝐵)) = (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵)
3 inss1 3979 . . . . . . 7 ((𝐴𝑉) ∩ 𝑉) ⊆ (𝐴𝑉)
4 ssid 3771 . . . . . . . 8 (𝐴𝑉) ⊆ (𝐴𝑉)
5 inss2 3980 . . . . . . . 8 (𝐴𝑉) ⊆ 𝑉
64, 5ssini 3982 . . . . . . 7 (𝐴𝑉) ⊆ ((𝐴𝑉) ∩ 𝑉)
73, 6eqssi 3766 . . . . . 6 ((𝐴𝑉) ∩ 𝑉) = (𝐴𝑉)
87difeq1i 3873 . . . . 5 (((𝐴𝑉) ∩ 𝑉) ∖ 𝐵) = ((𝐴𝑉) ∖ 𝐵)
92, 8eqtri 2792 . . . 4 ((𝐴𝑉) ∩ (𝑉𝐵)) = ((𝐴𝑉) ∖ 𝐵)
109eqeq1i 2775 . . 3 (((𝐴𝑉) ∩ (𝑉𝐵)) = (𝐴𝑉) ↔ ((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉))
11 eqcom 2777 . . 3 (((𝐴𝑉) ∖ 𝐵) = (𝐴𝑉) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
121, 10, 113bitri 286 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
13 disj3 4162 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ (𝐴𝑉) = ((𝐴𝑉) ∖ 𝐵))
14 in32 3972 . . 3 ((𝐴𝑉) ∩ 𝐵) = ((𝐴𝐵) ∩ 𝑉)
1514eqeq1i 2775 . 2 (((𝐴𝑉) ∩ 𝐵) = ∅ ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
1612, 13, 153bitr2i 288 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1630   ∖ cdif 3718   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062 This theorem is referenced by:  bj-sscon  33339
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