Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006)
Ref | Expression | ||
---|---|---|---|
Assertion | inabs | ⊢ ( 𝐴 ∩ ( 𝐴 ∪ 𝐵 ) ) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 | ⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) | |
2 | df-ss | ⊢ ( 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝐴 ∩ ( 𝐴 ∪ 𝐵 ) ) = 𝐴 ) | |
3 | 1 2 | mpbi | ⊢ ( 𝐴 ∩ ( 𝐴 ∪ 𝐵 ) ) = 𝐴 |