Theorem undifabs

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

	Ref	Expression
Assertion	undifabs	⊢ ( 𝐴 ∪ ( 𝐴 ∖ 𝐵 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		undif3	⊢ ( 𝐴 ∪ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝐴 ∪ 𝐴 ) ∖ ( 𝐵 ∖ 𝐴 ) )
2		unidm	⊢ ( 𝐴 ∪ 𝐴 ) = 𝐴
3	2	difeq1i	⊢ ( ( 𝐴 ∪ 𝐴 ) ∖ ( 𝐵 ∖ 𝐴 ) ) = ( 𝐴 ∖ ( 𝐵 ∖ 𝐴 ) )
4		difdif	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴
5	1 3 4	3eqtri	⊢ ( 𝐴 ∪ ( 𝐴 ∖ 𝐵 ) ) = 𝐴