Theorem undifabs

Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013)

	Ref	Expression
Assertion	undifabs	$$⊢A\cup \left(A\setminus B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		undif3	$$⊢A\cup \left(A\setminus B\right)=\left(A\cup A\right)\setminus \left(B\setminus A\right)$$
2		unidm	$$⊢A\cup A=A$$
3	2	difeq1i	$$⊢\left(A\cup A\right)\setminus \left(B\setminus A\right)=A\setminus \left(B\setminus A\right)$$
4		difdif	$$⊢A\setminus \left(B\setminus A\right)=A$$
5	1 3 4	3eqtri	$$⊢A\cup \left(A\setminus B\right)=A$$