Theorem brdif 5148

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 3908	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5096	. 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆))
3		df-br 5096	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5096	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	4	notbii 320	. . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
6	3, 5	anbi12i 628	. 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
7	1, 2, 6	3bitr4i 303	1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ∖ cdif 3895  ⟨cop 4583   class class class wbr 5095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-br 5096

This theorem is referenced by:  fundif  6538  fndmdif  6984  isocnv3  7275  brdifun  8661  dflt2  13053  pltval  18244  slenlt  27711  ltgov  28595  opeldifid  32600  qtophaus  33921  dftr6  35867  dffr5  35870  fundmpss  35883  brsset  36003  dfon3  36006  brtxpsd2  36009  dffun10  36028  elfuns  36029  dfrecs2  36066  dfrdg4  36067  dfint3  36068  brub  36070  broutsideof  36237  brvdif  38371  frege124d  43918