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Theorem disjdifb 48729
Description: Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
disjdifb ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅

Proof of Theorem disjdifb
StepHypRef Expression
1 indif1 4282 . 2 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐴 ∩ (𝐵𝐴)) ∖ 𝐵)
2 disjdif 4472 . . 3 (𝐴 ∩ (𝐵𝐴)) = ∅
32difeq1i 4122 . 2 ((𝐴 ∩ (𝐵𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4 0dif 4405 . 2 (∅ ∖ 𝐵) = ∅
51, 3, 43eqtri 2769 1 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cdif 3948  cin 3950  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:  iscnrm3r  48845
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