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Metamath Proof Explorer

Theorem tltnle

Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle . (Contributed by Thierry Arnoux, 11-Feb-2018)

Ref Expression
Hypotheses tleile.b 
|- B = ( Base ` K )
tleile.l 
|- .<_ = ( le ` K )
tltnle.s 
|- .< = ( lt ` K )
Assertion tltnle 
|- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> -. Y .<_ X ) )

Proof

Step Hyp Ref Expression
1 tleile.b 
 |-  B = ( Base ` K )
2 tleile.l 
 |-  .<_ = ( le ` K )
3 tltnle.s 
 |-  .< = ( lt ` K )
4 tospos 
 |-  ( K e. Toset -> K e. Poset )
5 1 2 3 pltval3 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )
6 4 5 syl3an1 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )
7 1 2 tleile 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .<_ X ) )
8 ibar 
 |-  ( ( X .<_ Y \/ Y .<_ X ) -> ( -. Y .<_ X <-> ( ( X .<_ Y \/ Y .<_ X ) /\ -. Y .<_ X ) ) )
9 pm5.61 
 |-  ( ( ( X .<_ Y \/ Y .<_ X ) /\ -. Y .<_ X ) <-> ( X .<_ Y /\ -. Y .<_ X ) )
10 8 9 bitr2di 
 |-  ( ( X .<_ Y \/ Y .<_ X ) -> ( ( X .<_ Y /\ -. Y .<_ X ) <-> -. Y .<_ X ) )
11 7 10 syl 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ -. Y .<_ X ) <-> -. Y .<_ X ) )
12 6 11 bitrd 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> -. Y .<_ X ) )