Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrnbtwn4 Unicode version

Theorem cvrnbtwn4 29762
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 23745 analog.) (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrle.b  |-  B  =  ( Base `  K
)
cvrle.l  |-  .<_  =  ( le `  K )
cvrle.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnbtwn4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<_  Y )  <->  ( X  =  Z  \/  Z  =  Y ) ) )

Proof of Theorem cvrnbtwn4
StepHypRef Expression
1 cvrle.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2404 . . . 4  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . . 4  |-  C  =  (  <o  `  K )
41, 2, 3cvrnbtwn 29754 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  -.  ( X
( lt `  K
) Z  /\  Z
( lt `  K
) Y ) )
5 iman 414 . . . . 5  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  -> 
( X  =  Z  \/  Z  =  Y ) )  <->  -.  (
( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) ) )
6 cvrle.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
76, 2pltval 14372 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X ( lt `  K ) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
873adant3r2 1163 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X
( lt `  K
) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
96, 2pltval 14372 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
1093com23 1159 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
11103adant3r1 1162 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Z
( lt `  K
) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
128, 11anbi12d 692 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X ( lt `  K ) Z  /\  Z ( lt `  K ) Y )  <-> 
( ( X  .<_  Z  /\  X  =/=  Z
)  /\  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) ) )
13 neanior 2652 . . . . . . . . 9  |-  ( ( X  =/=  Z  /\  Z  =/=  Y )  <->  -.  ( X  =  Z  \/  Z  =  Y )
)
1413anbi2i 676 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y
) ) )
15 an4 798 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1614, 15bitr3i 243 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1712, 16syl6rbbr 256 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( (
( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) )  <->  ( X
( lt `  K
) Z  /\  Z
( lt `  K
) Y ) ) )
1817notbid 286 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( -.  ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) )  <->  -.  ( X ( lt `  K ) Z  /\  Z ( lt `  K ) Y ) ) )
195, 18syl5rbb 250 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( -.  ( X ( lt `  K ) Z  /\  Z ( lt `  K ) Y )  <-> 
( ( X  .<_  Z  /\  Z  .<_  Y )  ->  ( X  =  Z  \/  Z  =  Y ) ) ) )
20193adant3 977 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( -.  ( X ( lt `  K ) Z  /\  Z ( lt `  K ) Y )  <-> 
( ( X  .<_  Z  /\  Z  .<_  Y )  ->  ( X  =  Z  \/  Z  =  Y ) ) ) )
214, 20mpbid 202 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<_  Y )  ->  ( X  =  Z  \/  Z  =  Y ) ) )
221, 6posref 14363 . . . . . . 7  |-  ( ( K  e.  Poset  /\  Z  e.  B )  ->  Z  .<_  Z )
23223ad2antr3 1124 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  .<_  Z )
24233adant3 977 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Z  .<_  Z )
25 breq1 4175 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Z  <->  Z  .<_  Z ) )
2624, 25syl5ibrcom 214 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
271, 6, 3cvrle 29761 . . . . . . . 8  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<_  Y )
2827ex 424 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  X  .<_  Y ) )
29283adant3r3 1164 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X C Y  ->  X  .<_  Y ) )
30293impia 1150 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<_  Y )
31 breq2 4176 . . . . 5  |-  ( Z  =  Y  ->  ( X  .<_  Z  <->  X  .<_  Y ) )
3230, 31syl5ibrcom 214 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  X  .<_  Z ) )
3326, 32jaod 370 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  X  .<_  Z ) )
34 breq1 4175 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Y  <->  Z  .<_  Y ) )
3530, 34syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<_  Y ) )
36 breq2 4176 . . . . 5  |-  ( Z  =  Y  ->  ( Z  .<_  Z  <->  Z  .<_  Y ) )
3724, 36syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  Z  .<_  Y ) )
3835, 37jaod 370 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  Z  .<_  Y ) )
3933, 38jcad 520 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  ( X  .<_  Z  /\  Z  .<_  Y ) ) )
4021, 39impbid 184 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<_  Y )  <->  ( X  =  Z  \/  Z  =  Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  cvrcmp  29766  leatb  29775  2llnmat  30006  2lnat  30266
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-poset 14358  df-plt 14370  df-covers 29749
  Copyright terms: Public domain W3C validator