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Theorem inindif 28738
 Description: See inundif 3998. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 3796 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 404 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 3804 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 3901 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 219 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  resf1o  28893  gsummptres  29115  measunl  29606  carsgclctun  29710  probdif  29809
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