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Theorem bj-sscon 33345
Description: Contraposition law for relative subsets. Relative and generalized version of ssconb 3894, which it can shorten, as well as conss2 39172. (Contributed by BJ, 11-Nov-2021.)
Assertion
Ref Expression
bj-sscon ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))

Proof of Theorem bj-sscon
StepHypRef Expression
1 incom 3956 . . . 4 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3961 . . 3 ((𝐴𝐵) ∩ 𝑉) = ((𝐵𝐴) ∩ 𝑉)
32eqeq1i 2776 . 2 (((𝐴𝐵) ∩ 𝑉) = ∅ ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
4 bj-disj2r 33344 . 2 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ ((𝐴𝐵) ∩ 𝑉) = ∅)
5 bj-disj2r 33344 . 2 ((𝐵𝑉) ⊆ (𝑉𝐴) ↔ ((𝐵𝐴) ∩ 𝑉) = ∅)
63, 4, 53bitr4i 292 1 ((𝐴𝑉) ⊆ (𝑉𝐵) ↔ (𝐵𝑉) ⊆ (𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  cdif 3720  cin 3722  wss 3723  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064
This theorem is referenced by: (None)
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