Theorem brdif 5163

Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)

Assertion

Ref	Expression
brdif	⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Proof of Theorem brdif

Step	Hyp	Ref	Expression
1		eldif 3923	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5111	. 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆))
3		df-br 5111	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5111	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	4	notbii 319	. . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
6	3, 5	anbi12i 627	. 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
7	1, 2, 6	3bitr4i 302	1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∖ cdif 3910  ⟨cop 4597   class class class wbr 5110

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-dif 3916  df-br 5111

This theorem is referenced by:  fundif  6555  fndmdif  6997  isocnv3  7282  brdifun  8684  dflt2  13077  pltval  18235  slenlt  27137  ltgov  27602  opeldifid  31584  qtophaus  32506  dftr6  34410  dffr5  34413  fundmpss  34427  brsset  34550  dfon3  34553  brtxpsd2  34556  dffun10  34575  elfuns  34576  dfrecs2  34611  dfrdg4  34612  dfint3  34613  brub  34615  broutsideof  34782  brvdif  36794  frege124d  42155