Theorem sscon

Description: Contraposition law for subsets. Exercise 15 of TakeutiZaring p. 22. (Contributed by NM, 22-Mar-1998)

	Ref	Expression
Assertion	sscon	$$⊢A\subseteq B\to C\setminus B\subseteq C\setminus A$$

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	con3d	$$⊢A\subseteq B\to \left(\neg x\in B\to \neg x\in A\right)$$
3	2	anim2d	$$⊢A\subseteq B\to \left(\left(x\in C\wedge \neg x\in B\right)\to \left(x\in C\wedge \neg x\in A\right)\right)$$
4		eldif	$$⊢x\in \left(C\setminus B\right)\leftrightarrow \left(x\in C\wedge \neg x\in B\right)$$
5		eldif	$$⊢x\in \left(C\setminus A\right)\leftrightarrow \left(x\in C\wedge \neg x\in A\right)$$
6	3 4 5	3imtr4g	$$⊢A\subseteq B\to \left(x\in \left(C\setminus B\right)\to x\in \left(C\setminus A\right)\right)$$
7	6	ssrdv	$$⊢A\subseteq B\to C\setminus B\subseteq C\setminus A$$