Theorem dfdif2 1495

Description: Alternate definition of class difference.

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem dfdif2

Step	Hyp	Ref	Expression
1		df-dif 1489	. 2 ⊢ (A ∖ B) = {x∣(x ∈ A ∧ ¬ x ∈ B)}
2		df-rab 1208	. 2 ⊢ {x ∈ A∣ ¬ x ∈ B} = {x∣(x ∈ A ∧ ¬ x ∈ B)}
3	1, 2	eqtr4 1122	1 ⊢ (A ∖ B) = {x ∈ A∣ ¬ x ∈ B}

Syntax hints:  ¬ wn 1   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204   ∖ cdif 1484

This theorem is referenced by:  kmlem3 3582

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097  df-rab 1208  df-dif 1489

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