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Theorem brdif 4837
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 3731 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 4785 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
3 df-br 4785 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 4785 . . . 4 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
54notbii 309 . . 3 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
63, 5anbi12i 604 . 2 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
71, 2, 63bitr4i 292 1 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wcel 2144  cdif 3718  cop 4320   class class class wbr 4784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-br 4785
This theorem is referenced by:  fundif  6078  fndmdif  6464  isocnv3  6724  brdifun  7924  dflt2  12185  pltval  17167  ltgov  25712  opeldifid  29744  qtophaus  30237  dftr6  31972  dffr5  31975  fundmpss  31996  slenlt  32208  brsset  32327  dfon3  32330  brtxpsd2  32333  dffun10  32352  elfuns  32353  dfrecs2  32388  dfrdg4  32389  dfint3  32390  brub  32392  broutsideof  32559  brvdif  34361  frege124d  38572
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