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Theorem inindif 30587
Description: See inundif 4398. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 4149 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 865 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 4158 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 4289 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 233 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:  wo 847   = wceq 1543  cdif 3868  cin 3870  wss 3871  c0 4242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3415  df-dif 3874  df-in 3878  df-ss 3888  df-nul 4243
This theorem is referenced by:  resf1o  30790  gsummptres  31036  indsumin  31707  measunl  31901  carsgclctun  32005  probdif  32104  hgt750lemd  32345
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