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Theorem bj-sscon 33956
Description: Contraposition law for relative subsets. Relative and generalized version of ssconb 4037, which it can shorten, as well as conss2 40327. (Contributed by BJ, 11-Nov-2021.)
Assertion
Ref Expression
bj-sscon ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Proof of Theorem bj-sscon
StepHypRef Expression
1 incom 4101 . . . 4 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 4107 . . 3 ((𝐴𝐵) ∩ 𝑉) = ((𝐵𝐴) ∩ 𝑉)
32eqeq1i 2799 . 2 (((𝐴𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
4 bj-disj2r 33955 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
5 bj-disj2r 33955 . 2 ((𝐵𝑉) ⊆ (𝑉𝐴) ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
63, 4, 53bitr4i 304 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1522  cdif 3858  cin 3860  wss 3861  c0 4213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rab 3113  df-v 3438  df-dif 3864  df-in 3868  df-ss 3876  df-nul 4214
This theorem is referenced by: (None)
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