Theorem dfdif2 3223

Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
dfdif2	⊢ (A ∖ B) = {x ∈ A ∣ ¬ x ∈ B}

Distinct variable groups:   x,A   x,B

Proof of Theorem dfdif2

Step	Hyp	Ref	Expression
1		eldif 3222	. . 3 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
2	1	abbi2i 2465	. 2 ⊢ (A ∖ B) = {x ∣ (x ∈ A ∧ ¬ x ∈ B)}
3		df-rab 2624	. 2 ⊢ {x ∈ A ∣ ¬ x ∈ B} = {x ∣ (x ∈ A ∧ ¬ x ∈ B)}
4	2, 3	eqtr4i 2376	1 ⊢ (A ∖ B) = {x ∈ A ∣ ¬ x ∈ B}

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619   ∖ cdif 3207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by:  difidALT  3620