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Metamath Proof Explorer

Theorem tlt3

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

Ref Expression
Hypotheses tlt3.b 
|- B = ( Base ` K )
tlt3.l 
|- .< = ( lt ` K )
Assertion tlt3 
|- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X = Y \/ X .< Y \/ Y .< X ) )

Proof

Step Hyp Ref Expression
1 tlt3.b 
 |-  B = ( Base ` K )
2 tlt3.l 
 |-  .< = ( lt ` K )
3 eqid 
 |-  ( le ` K ) = ( le ` K )
4 1 3 2 tlt2 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y \/ Y .< X ) )
5 tospos 
 |-  ( K e. Toset -> K e. Poset )
6 1 3 2 pleval2 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y <-> ( X .< Y \/ X = Y ) ) )
7 orcom 
 |-  ( ( X .< Y \/ X = Y ) <-> ( X = Y \/ X .< Y ) )
8 6 7 bitrdi 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y <-> ( X = Y \/ X .< Y ) ) )
9 5 8 syl3an1 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y <-> ( X = Y \/ X .< Y ) ) )
10 9 orbi1d 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y \/ Y .< X ) <-> ( ( X = Y \/ X .< Y ) \/ Y .< X ) ) )
11 4 10 mpbid 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X = Y \/ X .< Y ) \/ Y .< X ) )
12 df-3or 
 |-  ( ( X = Y \/ X .< Y \/ Y .< X ) <-> ( ( X = Y \/ X .< Y ) \/ Y .< X ) )
13 11 12 sylibr 
 |-  ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X = Y \/ X .< Y \/ Y .< X ) )