Theorem conss2

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

	Ref	Expression
Assertion	conss2	$$⊢A\subseteq \mathrm{V}\setminus B\leftrightarrow B\subseteq \mathrm{V}\setminus A$$

Proof

Step	Hyp	Ref	Expression
1		ssv	$$⊢A\subseteq \mathrm{V}$$
2		ssv	$$⊢B\subseteq \mathrm{V}$$
3		ssconb	$$⊢\left(A\subseteq \mathrm{V}\wedge B\subseteq \mathrm{V}\right)\to \left(A\subseteq \mathrm{V}\setminus B\leftrightarrow B\subseteq \mathrm{V}\setminus A\right)$$
4	1 2 3	mp2an	$$⊢A\subseteq \mathrm{V}\setminus B\leftrightarrow B\subseteq \mathrm{V}\setminus A$$