Theorem ssindif0

Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003)

	Ref	Expression
Assertion	ssindif0	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ ( V ∖ 𝐵 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		disj2	⊢ ( ( 𝐴 ∩ ( V ∖ 𝐵 ) ) = ∅ ↔ 𝐴 ⊆ ( V ∖ ( V ∖ 𝐵 ) ) )
2		ddif	⊢ ( V ∖ ( V ∖ 𝐵 ) ) = 𝐵
3	2	sseq2i	⊢ ( 𝐴 ⊆ ( V ∖ ( V ∖ 𝐵 ) ) ↔ 𝐴 ⊆ 𝐵 )
4	1 3	bitr2i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ ( V ∖ 𝐵 ) ) = ∅ )