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Theorem cvrnbtwn 30143
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 23794 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 30141 . . . 4  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
543adant3r3 1165 . . 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 2717 . . . . . . 7  |-  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )
7 breq2 4219 . . . . . . . . . 10  |-  ( z  =  Z  ->  ( X  .<  z  <->  X  .<  Z ) )
8 breq1 4218 . . . . . . . . . 10  |-  ( z  =  Z  ->  (
z  .<  Y  <->  Z  .<  Y ) )
97, 8anbi12d 693 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( X  .<  z  /\  z  .<  Y )  <-> 
( X  .<  Z  /\  Z  .<  Y ) ) )
109notbid 287 . . . . . . . 8  |-  ( z  =  Z  ->  ( -.  ( X  .<  z  /\  z  .<  Y )  <->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1110rspcv 3050 . . . . . . 7  |-  ( Z  e.  B  ->  ( A. z  e.  B  -.  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
126, 11syl5bir 211 . . . . . 6  |-  ( Z  e.  B  ->  ( -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
1312adantld 455 . . . . 5  |-  ( Z  e.  B  ->  (
( X  .<  Y  /\  -.  E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
14133ad2ant3 981 . . . 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 454 . . 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 208 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) ) )
17163impia 1151 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215   ` cfv 5457   Basecbs 13474   ltcplt 14403    <o ccvr 30134
This theorem is referenced by:  cvrnbtwn2  30147  cvrnbtwn3  30148  cvrnbtwn4  30151  ltltncvr  30294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-covers 30138
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