Theorem undifr

Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023)

	Ref	Expression
Assertion	undifr	$$⊢A\subseteq B\leftrightarrow \left(B\setminus A\right)\cup A=B$$

Proof

Step	Hyp	Ref	Expression
1		undif	$$⊢A\subseteq B\leftrightarrow A\cup \left(B\setminus A\right)=B$$
2		uncom	$$⊢A\cup \left(B\setminus A\right)=\left(B\setminus A\right)\cup A$$
3	2	eqeq1i	$$⊢A\cup \left(B\setminus A\right)=B\leftrightarrow \left(B\setminus A\right)\cup A=B$$
4	1 3	bitri	$$⊢A\subseteq B\leftrightarrow \left(B\setminus A\right)\cup A=B$$