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Theorem inindif 4375
Description: The intersection and class difference of a class with another class are disjoint. With inundif 4479, 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 4238 . . 3 (𝐴𝐶) ⊆ 𝐶
2 ssinss1 4246 . . 3 ((𝐴𝐶) ⊆ 𝐶 → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
31, 2ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
4 inssdif0 4374 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
53, 4mpbi 230 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cdif 3948  cin 3950  wss 3951  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334
This theorem is referenced by:  resf1o  32741  indsumin  32847  gsummptres  33055  measunl  34217  carsgclctun  34323  probdif  34422  hgt750lemd  34663  redvmptabs  42390
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