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Theorem bj-sscon 34780
 Description: Contraposition law for relative subclasses. Relative and generalized version of ssconb 4045, which it can shorten, as well as conss2 41555. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4045 nor conss2 41555. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sscon ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Proof of Theorem bj-sscon
StepHypRef Expression
1 incom 4108 . . . 4 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 4115 . . 3 ((𝐴𝐵) ∩ 𝑉) = ((𝐵𝐴) ∩ 𝑉)
32eqeq1i 2763 . 2 (((𝐴𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
4 bj-disj2r 34779 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
5 bj-disj2r 34779 . 2 ((𝐵𝑉) ⊆ (𝑉𝐴) ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
63, 4, 53bitr4i 306 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3863  df-in 3867  df-ss 3877  df-nul 4228 This theorem is referenced by: (None)
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