Metamath Proof Explorer


Theorem chabs2i

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 16-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses chabs.1 𝐴C
chabs.2 𝐵C
Assertion chabs2i ( 𝐴 ∩ ( 𝐴 𝐵 ) ) = 𝐴

Proof

Step Hyp Ref Expression
1 chabs.1 𝐴C
2 chabs.2 𝐵C
3 chabs2 ( ( 𝐴C𝐵C ) → ( 𝐴 ∩ ( 𝐴 𝐵 ) ) = 𝐴 )
4 1 2 3 mp2an ( 𝐴 ∩ ( 𝐴 𝐵 ) ) = 𝐴