Theorem difun2

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

	Ref	Expression
Assertion	difun2	⊢ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = ( 𝐴 ∖ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		difundir	⊢ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐵 ) )
2		difid	⊢ ( 𝐵 ∖ 𝐵 ) = ∅
3	2	uneq2i	⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐵 ∖ 𝐵 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ∅ )
4		un0	⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ ∅ ) = ( 𝐴 ∖ 𝐵 )
5	1 3 4	3eqtri	⊢ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = ( 𝐴 ∖ 𝐵 )