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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 )