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Theorem bj-sscon 36214
Description: Contraposition law for relative subclasses. Relative and generalized version of ssconb 4138, which it can shorten, as well as conss2 43505. (Contributed by BJ, 11-Nov-2021.) This proof does not rely, even indirectly, on ssconb 4138 nor conss2 43505. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sscon ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Proof of Theorem bj-sscon
StepHypRef Expression
1 incom 4202 . . . 4 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 4209 . . 3 ((𝐴𝐵) ∩ 𝑉) = ((𝐵𝐴) ∩ 𝑉)
32eqeq1i 2736 . 2 (((𝐴𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
4 bj-disj2r 36213 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
5 bj-disj2r 36213 . 2 ((𝐵𝑉) ⊆ (𝑉𝐴) ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
63, 4, 53bitr4i 302 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  cdif 3946  cin 3948  wss 3949  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324
This theorem is referenced by: (None)
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