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Theorem inindif 4381
Description: The intersection and class difference of a class with another class are disjoint. With inundif 4485, this shows that such intersection and class difference partition the class 𝐴. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 4246 . . 3 (𝐴𝐶) ⊆ 𝐶
2 ssinss1 4254 . . 3 ((𝐴𝐶) ⊆ 𝐶 → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
31, 2ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
4 inssdif0 4380 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
53, 4mpbi 230 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cin 3962  wss 3963  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  resf1o  32748  gsummptres  33038  indsumin  34003  measunl  34197  carsgclctun  34303  probdif  34402  hgt750lemd  34642  redvmptabs  42369
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