ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brdif Structured version   GIF version

Theorem brdif 3782
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 2900 . 2 (⟨A, B (𝑅𝑆) ↔ (⟨A, B 𝑅 ¬ ⟨A, B 𝑆))
2 df-br 3735 . 2 (A(𝑅𝑆)B ↔ ⟨A, B (𝑅𝑆))
3 df-br 3735 . . 3 (A𝑅B ↔ ⟨A, B 𝑅)
4 df-br 3735 . . . 4 (A𝑆B ↔ ⟨A, B 𝑆)
54notbii 581 . . 3 A𝑆B ↔ ¬ ⟨A, B 𝑆)
63, 5anbi12i 436 . 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 1370  cdif 2887  cop 3349   class class class wbr 3734
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-br 3735
This theorem is referenced by:  fndmdif  5193  brdifun  6040
  Copyright terms: Public domain W3C validator