Theorem conss2

Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011)

	Ref	Expression
Assertion	conss2	⊢ ( 𝐴 ⊆ ( V ∖ 𝐵 ) ↔ 𝐵 ⊆ ( V ∖ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐴 ⊆ V
2		ssv	⊢ 𝐵 ⊆ V
3		ssconb	⊢ ( ( 𝐴 ⊆ V ∧ 𝐵 ⊆ V ) → ( 𝐴 ⊆ ( V ∖ 𝐵 ) ↔ 𝐵 ⊆ ( V ∖ 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ⊆ ( V ∖ 𝐵 ) ↔ 𝐵 ⊆ ( V ∖ 𝐴 ) )