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Metamath Proof Explorer

Theorem atn0

Description: An atom is not zero. ( atne0 analog.) (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses atne0.z 
|- .0. = ( 0. ` K )
atne0.a 
|- A = ( Atoms ` K )
Assertion atn0 
|- ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )

Proof

Step Hyp Ref Expression
1 atne0.z 
 |-  .0. = ( 0. ` K )
2 atne0.a 
 |-  A = ( Atoms ` K )
3 eqid 
 |-  ( Base ` K ) = ( Base ` K )
4 eqid 
 |-  ( le ` K ) = ( le ` K )
5 3 4 1 2 isat3 
 |-  ( K e. AtLat -> ( P e. A <-> ( P e. ( Base ` K ) /\ P =/= .0. /\ A. x e. ( Base ` K ) ( x ( le ` K ) P -> ( x = P \/ x = .0. ) ) ) ) )
6 simp2 
 |-  ( ( P e. ( Base ` K ) /\ P =/= .0. /\ A. x e. ( Base ` K ) ( x ( le ` K ) P -> ( x = P \/ x = .0. ) ) ) -> P =/= .0. )
7 5 6 biimtrdi 
 |-  ( K e. AtLat -> ( P e. A -> P =/= .0. ) )
8 7 imp 
 |-  ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )