Theorem dfdif2

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

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

Proof

Step	Hyp	Ref	Expression
1		df-dif	⊢ ( 𝐴 ∖ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) }
2		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) }
3	1 2	eqtr4i	⊢ ( 𝐴 ∖ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵 }