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Metamath Proof Explorer

Theorem cvrnbtwn

Description: There is no element between the two arguments of the covers relation. ( cvnbtwn analog.) (Contributed by NM, 18-Oct-2011)

Ref Expression
Hypotheses cvrfval.b 
|- B = ( Base ` K )
cvrfval.s 
|- .< = ( lt ` K )
cvrfval.c 
|- C = (
Assertion cvrnbtwn 
|- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X .< Z /\ Z .< Y ) )

Proof

Step Hyp Ref Expression
1 cvrfval.b 
 |-  B = ( Base ` K )
2 cvrfval.s 
 |-  .< = ( lt ` K )
3 cvrfval.c 
 |-  C = (
4 1 2 3 cvrval 
 |-  ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
5 4 3adant3r3 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
6 ralnex 
 |-  ( A. z e. B -. ( X .< z /\ z .< Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) )
7 breq2 
 |-  ( z = Z -> ( X .< z <-> X .< Z ) )
8 breq1 
 |-  ( z = Z -> ( z .< Y <-> Z .< Y ) )
9 7 8 anbi12d 
 |-  ( z = Z -> ( ( X .< z /\ z .< Y ) <-> ( X .< Z /\ Z .< Y ) ) )
10 9 notbid 
 |-  ( z = Z -> ( -. ( X .< z /\ z .< Y ) <-> -. ( X .< Z /\ Z .< Y ) ) )
11 10 rspcv 
 |-  ( Z e. B -> ( A. z e. B -. ( X .< z /\ z .< Y ) -> -. ( X .< Z /\ Z .< Y ) ) )
12 6 11 biimtrrid 
 |-  ( Z e. B -> ( -. E. z e. B ( X .< z /\ z .< Y ) -> -. ( X .< Z /\ Z .< Y ) ) )
13 12 adantld 
 |-  ( Z e. B -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
14 13 3ad2ant3 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
15 14 adantl 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
16 5 15 sylbid 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y -> -. ( X .< Z /\ Z .< Y ) ) )
17 16 3impia 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X .< Z /\ Z .< Y ) )