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Metamath Proof Explorer

Theorem pltval3

Description: Alternate expression for the "less than" relation. ( dfpss3 analog.) (Contributed by NM, 4-Nov-2011)

Ref Expression
Hypotheses pleval2.b 
|- B = ( Base ` K )
pleval2.l 
|- .<_ = ( le ` K )
pleval2.s 
|- .< = ( lt ` K )
Assertion pltval3 
|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )

Proof

Step Hyp Ref Expression
1 pleval2.b 
 |-  B = ( Base ` K )
2 pleval2.l 
 |-  .<_ = ( le ` K )
3 pleval2.s 
 |-  .< = ( lt ` K )
4 2 3 pltval 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) )
5 1 2 posref 
 |-  ( ( K e. Poset /\ X e. B ) -> X .<_ X )
6 5 3adant3 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> X .<_ X )
7 breq1 
 |-  ( X = Y -> ( X .<_ X <-> Y .<_ X ) )
8 6 7 syl5ibcom 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X = Y -> Y .<_ X ) )
9 8 adantr 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X = Y -> Y .<_ X ) )
10 1 2 posasymb 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
11 10 biimpd 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) )
12 11 expdimp 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( Y .<_ X -> X = Y ) )
13 9 12 impbid 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X = Y <-> Y .<_ X ) )
14 13 necon3abid 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X =/= Y <-> -. Y .<_ X ) )
15 14 pm5.32da 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ X =/= Y ) <-> ( X .<_ Y /\ -. Y .<_ X ) ) )
16 4 15 bitrd 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ -. Y .<_ X ) ) )