Theorem ssdif

Description: Difference law for subsets. (Contributed by NM, 28-May-1998)

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

Proof

Step	Hyp	Ref	Expression
1		ssel	$$⊢A\subseteq B\to \left(x\in A\to x\in B\right)$$
2	1	anim1d	$$⊢A\subseteq B\to \left(\left(x\in A\wedge \neg x\in C\right)\to \left(x\in B\wedge \neg x\in C\right)\right)$$
3		eldif	$$⊢x\in \left(A\setminus C\right)\leftrightarrow \left(x\in A\wedge \neg x\in C\right)$$
4		eldif	$$⊢x\in \left(B\setminus C\right)\leftrightarrow \left(x\in B\wedge \neg x\in C\right)$$
5	2 3 4	3imtr4g	$$⊢A\subseteq B\to \left(x\in \left(A\setminus C\right)\to x\in \left(B\setminus C\right)\right)$$
6	5	ssrdv	$$⊢A\subseteq B\to A\setminus C\subseteq B\setminus C$$