Metamath Proof Explorer


Theorem leat2

Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013)

Ref Expression
Hypotheses leatom.b
|- B = ( Base ` K )
leatom.l
|- .<_ = ( le ` K )
leatom.z
|- .0. = ( 0. ` K )
leatom.a
|- A = ( Atoms ` K )
Assertion leat2
|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ ( X =/= .0. /\ X .<_ P ) ) -> X = P )

Proof

Step Hyp Ref Expression
1 leatom.b
 |-  B = ( Base ` K )
2 leatom.l
 |-  .<_ = ( le ` K )
3 leatom.z
 |-  .0. = ( 0. ` K )
4 leatom.a
 |-  A = ( Atoms ` K )
5 1 2 3 4 leatb
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X = P \/ X = .0. ) ) )
6 orcom
 |-  ( ( X = P \/ X = .0. ) <-> ( X = .0. \/ X = P ) )
7 neor
 |-  ( ( X = .0. \/ X = P ) <-> ( X =/= .0. -> X = P ) )
8 6 7 bitri
 |-  ( ( X = P \/ X = .0. ) <-> ( X =/= .0. -> X = P ) )
9 5 8 bitrdi
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X =/= .0. -> X = P ) ) )
10 9 biimpd
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P -> ( X =/= .0. -> X = P ) ) )
11 10 com23
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X =/= .0. -> ( X .<_ P -> X = P ) ) )
12 11 imp32
 |-  ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ ( X =/= .0. /\ X .<_ P ) ) -> X = P )