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Theorem disjresdif 38197
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 38196 . 2 ((𝐴𝐵) = ∅ → ((𝑅𝐴) ∩ (𝑅𝐵)) = ∅)
2 disjdif2 4503 . 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 1537  cdif 3973  cin 3975  c0 4352  cres 5702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707  df-res 5712
This theorem is referenced by:  disjresundif  38198
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