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Metamath Proof Explorer

Theorem tlt2

Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018)

Ref Expression
Hypotheses tlt2.b 
|- B = ( Base ` K )
tlt2.e 
|- .<_ = ( le ` K )
tlt2.l 
|- .< = ( lt ` K )
Assertion tlt2 
|- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .< X ) )

Proof

Step Hyp Ref Expression
1 tlt2.b 
 |-  B = ( Base ` K )
2 tlt2.e 
 |-  .<_ = ( le ` K )
3 tlt2.l 
 |-  .< = ( lt ` K )
4 exmidd 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ -. X .<_ Y ) )
5 1 2 3 tltnle 
 |-  ( ( K e. Toset /\ Y e. B /\ X e. B ) -> ( Y .< X <-> -. X .<_ Y ) )
6 5 3com23 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( Y .< X <-> -. X .<_ Y ) )
7 6 orbi2d 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y \/ Y .< X ) <-> ( X .<_ Y \/ -. X .<_ Y ) ) )
8 4 7 mpbird 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .< X ) )