Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjimin Structured version   Visualization version   GIF version

Theorem disjimin 36051
 Description: Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.)
Assertion
Ref Expression
disjimin ( Disj 𝑆 → Disj (𝑅𝑆))

Proof of Theorem disjimin
StepHypRef Expression
1 inss2 4190 . 2 (𝑅𝑆) ⊆ 𝑆
21disjssi 36035 1 ( Disj 𝑆 → Disj (𝑅𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3918   Disj wdisjALTV 35557 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-coss 35729  df-cnvrefrel 35835  df-funALTV 35985  df-disjALTV 36008 This theorem is referenced by:  disjiminres  36052
 Copyright terms: Public domain W3C validator