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Theorem inindif 32305
Description: See inundif 4474. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 4225 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 864 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 4234 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 4365 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 229 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:  wo 846   = wceq 1534  cdif 3942  cin 3944  wss 3945  c0 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-ss 3962  df-nul 4319
This theorem is referenced by:  resf1o  32506  gsummptres  32760  indsumin  33635  measunl  33829  carsgclctun  33935  probdif  34034  hgt750lemd  34274
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