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Theorem bj-sscon 33462
Description: Contraposition law for relative subsets. Relative and generalized version of ssconb 3907, which it can shorten, as well as conss2 39343. (Contributed by BJ, 11-Nov-2021.)
Assertion
Ref Expression
bj-sscon ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Proof of Theorem bj-sscon
StepHypRef Expression
1 incom 3969 . . . 4 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3974 . . 3 ((𝐴𝐵) ∩ 𝑉) = ((𝐵𝐴) ∩ 𝑉)
32eqeq1i 2770 . 2 (((𝐴𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
4 bj-disj2r 33461 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
5 bj-disj2r 33461 . 2 ((𝐵𝑉) ⊆ (𝑉𝐴) ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
63, 4, 53bitr4i 294 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  cdif 3731  cin 3733  wss 3734  c0 4081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3737  df-in 3741  df-ss 3748  df-nul 4082
This theorem is referenced by: (None)
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