Metamath Proof Explorer


Theorem inabs

Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006)

Ref Expression
Assertion inabs ( 𝐴 ∩ ( 𝐴𝐵 ) ) = 𝐴

Proof

Step Hyp Ref Expression
1 ssun1 𝐴 ⊆ ( 𝐴𝐵 )
2 df-ss ( 𝐴 ⊆ ( 𝐴𝐵 ) ↔ ( 𝐴 ∩ ( 𝐴𝐵 ) ) = 𝐴 )
3 1 2 mpbi ( 𝐴 ∩ ( 𝐴𝐵 ) ) = 𝐴