Theorem difcom

Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007)

	Ref	Expression
Assertion	difcom	$$⊢A\setminus B\subseteq C\leftrightarrow A\setminus C\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		uncom	$$⊢B\cup C=C\cup B$$
2	1	sseq2i	$$⊢A\subseteq B\cup C\leftrightarrow A\subseteq C\cup B$$
3		ssundif	$$⊢A\subseteq B\cup C\leftrightarrow A\setminus B\subseteq C$$
4		ssundif	$$⊢A\subseteq C\cup B\leftrightarrow A\setminus C\subseteq B$$
5	2 3 4	3bitr3i	$$⊢A\setminus B\subseteq C\leftrightarrow A\setminus C\subseteq B$$