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Theorem disjresdif 37769
Description: The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024.)
Assertion
Ref Expression
disjresdif ((𝐴𝐵) = ∅ → ((𝑅𝐴) ∖ (𝑅𝐵)) = (𝑅𝐴))

Proof of Theorem disjresdif
StepHypRef Expression
1 disjresdisj 37768 . 2 ((𝐴𝐵) = ∅ → ((𝑅𝐴) ∩ (𝑅𝐵)) = ∅)
2 disjdif2 4475 . 2 (((𝑅𝐴) ∩ (𝑅𝐵)) = ∅ → ((𝑅𝐴) ∖ (𝑅𝐵)) = (𝑅𝐴))
31, 2syl 17 1 ((𝐴𝐵) = ∅ → ((𝑅𝐴) ∖ (𝑅𝐵)) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cdif 3936  cin 3938  c0 4318  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5206  df-xp 5678  df-rel 5679  df-res 5684
This theorem is referenced by:  disjresundif  37770
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