Theorem undifr

Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023) (Proof shortened by SN, 11-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ssequn2	$$⊢A\subseteq B\leftrightarrow B\cup A=B$$
2		undif1	$$⊢\left(B\setminus A\right)\cup A=B\cup A$$
3	2	eqeq1i	$$⊢\left(B\setminus A\right)\cup A=B\leftrightarrow B\cup A=B$$
4	1 3	bitr4i	$$⊢A\subseteq B\leftrightarrow \left(B\setminus A\right)\cup A=B$$