Theorem difexg

Description: Existence of a difference. (Contributed by NM, 26-May-1998)

	Ref	Expression
Assertion	difexg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		difss	⊢ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴
2		ssexg	⊢ ( ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∖ 𝐵 ) ∈ V )
3	1 2	mpan	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝐵 ) ∈ V )