Theorem brdif 5153

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 3913	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5101	. 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆))
3		df-br 5101	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5101	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	4	notbii 320	. . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
6	3, 5	anbi12i 629	. 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 2114   ∖ cdif 3900  ⟨cop 4588   class class class wbr 5100

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-br 5101

This theorem is referenced by:  fundif  6549  fndmdif  6996  isocnv3  7288  brdifun  8676  dflt2  13074  pltval  18265  lenlts  27732  ltgov  28681  opeldifid  32685  qtophaus  34013  dftr6  35964  dffr5  35967  fundmpss  35980  brsset  36100  dfon3  36103  brtxpsd2  36106  dffun10  36125  elfuns  36126  dfrecs2  36163  dfrdg4  36164  dfint3  36165  brub  36167  broutsideof  36334  brvdif  38514  frege124d  44114