MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brdif Structured version   Visualization version   GIF version

Theorem brdif 5095
Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
Assertion
Ref Expression
brdif (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Proof of Theorem brdif
StepHypRef Expression
1 eldif 3918 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 5043 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
3 df-br 5043 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 5043 . . . 4 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
54notbii 323 . . 3 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
63, 5anbi12i 629 . 2 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
71, 2, 63bitr4i 306 1 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wcel 2114  cdif 3905  cop 4545   class class class wbr 5042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-dif 3911  df-br 5043
This theorem is referenced by:  fundif  6382  fndmdif  6794  isocnv3  7069  brdifun  8305  dflt2  12529  pltval  17561  ltgov  26389  opeldifid  30357  qtophaus  31158  dftr6  33060  dffr5  33063  fundmpss  33083  slenlt  33305  brsset  33424  dfon3  33427  brtxpsd2  33430  dffun10  33449  elfuns  33450  dfrecs2  33485  dfrdg4  33486  dfint3  33487  brub  33489  broutsideof  33656  brvdif  35641  frege124d  40393
  Copyright terms: Public domain W3C validator