Theorem brdif 4694

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 3221	. 2 ⊢ (⟨A, B⟩ ∈ (R ∖ S) ↔ (⟨A, B⟩ ∈ R ∧ ¬ ⟨A, B⟩ ∈ S))
2		df-br 4640	. 2 ⊢ (A(R ∖ S)B ↔ ⟨A, B⟩ ∈ (R ∖ S))
3		df-br 4640	. . 3 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
4		df-br 4640	. . . 4 ⊢ (ASB ↔ ⟨A, B⟩ ∈ S)
5	4	notbii 287	. . 3 ⊢ (¬ ASB ↔ ¬ ⟨A, B⟩ ∈ S)
6	3, 5	anbi12i 678	. 2 ⊢ ((ARB ∧ ¬ ASB) ↔ (⟨A, B⟩ ∈ R ∧ ¬ ⟨A, B⟩ ∈ S))
7	1, 2, 6	3bitr4i 268	1 ⊢ (A(R ∖ S)B ↔ (ARB ∧ ¬ ASB))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3206  ⟨cop 4561   class class class wbr 4639

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-br 4640

This theorem is referenced by:  fnfullfunlem1  5856  brltc  6114