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Theorem brdif 3803
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 2921 . 2 (⟨A, B (𝑅𝑆) ↔ (⟨A, B 𝑅 ¬ ⟨A, B 𝑆))
2 df-br 3756 . 2 (A(𝑅𝑆)B ↔ ⟨A, B (𝑅𝑆))
3 df-br 3756 . . 3 (A𝑅B ↔ ⟨A, B 𝑅)
4 df-br 3756 . . . 4 (A𝑆B ↔ ⟨A, B 𝑆)
54notbii 593 . . 3 A𝑆B ↔ ¬ ⟨A, B 𝑆)
63, 5anbi12i 433 . 2 ((A𝑅B ¬ A𝑆B) ↔ (⟨A, B 𝑅 ¬ ⟨A, B 𝑆))
71, 2, 63bitr4i 201 1 (A(𝑅𝑆)B ↔ (A𝑅B ¬ A𝑆B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wcel 1390  cdif 2908  cop 3370   class class class wbr 3755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-br 3756
This theorem is referenced by:  fndmdif  5215  brdifun  6069
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