Theorem brdif 4083

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 3163	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 4031	. 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∖ 𝑆))
3		df-br 4031	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 4031	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	4	notbii 669	. . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
6	3, 5	anbi12i 460	. 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
7	1, 2, 6	3bitr4i 212	1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∈ wcel 2164   ∖ cdif 3151  ⟨cop 3622   class class class wbr 4030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-br 4031

This theorem is referenced by:  fndmdif  5664  brdifun  6616