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Theorem disjdifb 48657
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 4287 . 2 ((𝐴𝐵) ∩ (𝐵𝐴)) = ((𝐴 ∩ (𝐵𝐴)) ∖ 𝐵)
2 disjdif 4477 . . 3 (𝐴 ∩ (𝐵𝐴)) = ∅
32difeq1i 4131 . 2 ((𝐴 ∩ (𝐵𝐴)) ∖ 𝐵) = (∅ ∖ 𝐵)
4 0dif 4410 . 2 (∅ ∖ 𝐵) = ∅
51, 3, 43eqtri 2766 1 ((𝐴𝐵) ∩ (𝐵𝐴)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  cdif 3959  cin 3961  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-in 3969  df-ss 3979  df-nul 4339
This theorem is referenced by:  iscnrm3r  48744
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