Theorem undif2

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif ). Part of proof of Corollary 6K of Enderton p. 144. (Contributed by NM, 19-May-1998)

	Ref	Expression
Assertion	undif2	$$⊢A\cup \left(B\setminus A\right)=A\cup B$$

Proof

Step	Hyp	Ref	Expression
1		uncom	$$⊢A\cup \left(B\setminus A\right)=\left(B\setminus A\right)\cup A$$
2		undif1	$$⊢\left(B\setminus A\right)\cup A=B\cup A$$
3		uncom	$$⊢B\cup A=A\cup B$$
4	1 2 3	3eqtri	$$⊢A\cup \left(B\setminus A\right)=A\cup B$$