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Theorem cvrnbtwn 29754
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 23742 analog.) (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrfval.b  |-  B  =  ( Base `  K
)
cvrfval.s  |-  .<  =  ( lt `  K )
cvrfval.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnbtwn  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )

Proof of Theorem cvrnbtwn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cvrfval.b . . . . 5  |-  B  =  ( Base `  K
)
2 cvrfval.s . . . . 5  |-  .<  =  ( lt `  K )
3 cvrfval.c . . . . 5  |-  C  =  (  <o  `  K )
41, 2, 3cvrval 29752 . . . 4  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
543adant3r3 1164 . . 3  |-  ( ( 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 2676 . . . . . . 7  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
7 breq2 4176 . . . . . . . . . 10  |-  ( z  =  Z  ->  ( X  .<  z  <->  X  .<  Z ) )
8 breq1 4175 . . . . . . . . . 10  |-  ( z  =  Z  ->  (
z  .<  Y  <->  Z  .<  Y ) )
97, 8anbi12d 692 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  Z  /\  Z  .<  Y ) ) )
109notbid 286 . . . . . . . 8  |-  ( z  =  Z  ->  ( -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1110rspcv 3008 . . . . . . 7  |-  ( Z  e.  B  ->  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
126, 11syl5bir 210 . . . . . 6  |-  ( Z  e.  B  ->  ( -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1312adantld 454 . . . . 5  |-  ( Z  e.  B  ->  (
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
14133ad2ant3 980 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1514adantl 453 . . 3  |-  ( ( 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 ) ) )
165, 15sylbid 207 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
17163impia 1150 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  cvrnbtwn2  29758  cvrnbtwn3  29759  cvrnbtwn4  29762  ltltncvr  29905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-covers 29749
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