Theorem difun2

Description: Absorption of union by difference. Theorem 36 of Suppes p. 29. (Contributed by NM, 19-May-1998)

	Ref	Expression
Assertion	difun2	$$⊢\left(A\cup B\right)\setminus B=A\setminus B$$

Proof

Step	Hyp	Ref	Expression
1		difundir	$$⊢\left(A\cup B\right)\setminus B=\left(A\setminus B\right)\cup \left(B\setminus B\right)$$
2		difid	$$⊢B\setminus B=\varnothing$$
3	2	uneq2i	$$⊢\left(A\setminus B\right)\cup \left(B\setminus B\right)=\left(A\setminus B\right)\cup \varnothing$$
4		un0	$$⊢\left(A\setminus B\right)\cup \varnothing =A\setminus B$$
5	1 3 4	3eqtri	$$⊢\left(A\cup B\right)\setminus B=A\setminus B$$