Metamath Proof Explorer


Theorem isat2

Description: The predicate "is an atom". ( elatcv0 analog.) (Contributed by NM, 18-Jun-2012)

Ref Expression
Hypotheses isatom.b 𝐵 = ( Base ‘ 𝐾 )
isatom.z 0 = ( 0. ‘ 𝐾 )
isatom.c 𝐶 = ( ⋖ ‘ 𝐾 )
isatom.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion isat2 ( ( 𝐾𝐷𝑃𝐵 ) → ( 𝑃𝐴0 𝐶 𝑃 ) )

Proof

Step Hyp Ref Expression
1 isatom.b 𝐵 = ( Base ‘ 𝐾 )
2 isatom.z 0 = ( 0. ‘ 𝐾 )
3 isatom.c 𝐶 = ( ⋖ ‘ 𝐾 )
4 isatom.a 𝐴 = ( Atoms ‘ 𝐾 )
5 1 2 3 4 isat ( 𝐾𝐷 → ( 𝑃𝐴 ↔ ( 𝑃𝐵0 𝐶 𝑃 ) ) )
6 5 baibd ( ( 𝐾𝐷𝑃𝐵 ) → ( 𝑃𝐴0 𝐶 𝑃 ) )