Theorem brdif 4042

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 3130	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 3990	. 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆))
3		df-br 3990	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 3990	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	4	notbii 663	. . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
6	3, 5	anbi12i 457	. 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
7	1, 2, 6	3bitr4i 211	1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∈ wcel 2141   ∖ cdif 3118  ⟨cop 3586   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-br 3990

This theorem is referenced by:  fndmdif  5601  brdifun  6540