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Theorem cvrnbtwn4 29542
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22871 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 2285 . . . 4  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . . 4  |-  C  =  (  <o  `  K )
41, 2, 3cvrnbtwn 29534 . . 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 413 . . . . 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 14096 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X ( lt `  K ) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
873adant3r2 1161 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X
( lt `  K
) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
96, 2pltval 14096 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
1093com23 1157 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
11103adant3r1 1160 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Z
( lt `  K
) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
128, 11anbi12d 691 . . . . . . 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 2533 . . . . . . . . 9  |-  ( ( X  =/=  Z  /\  Z  =/=  Y )  <->  -.  ( X  =  Z  \/  Z  =  Y )
)
1413anbi2i 675 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y
) ) )
15 an4 797 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1614, 15bitr3i 242 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1712, 16syl6rbbr 255 . . . . . 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 285 . . . . 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 249 . . . 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 975 . . 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 201 . 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 14087 . . . . . . 7  |-  ( ( K  e.  Poset  /\  Z  e.  B )  ->  Z  .<_  Z )
23223ad2antr3 1122 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  .<_  Z )
24233adant3 975 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Z  .<_  Z )
25 breq1 4028 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Z  <->  Z  .<_  Z ) )
2624, 25syl5ibrcom 213 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
271, 6, 3cvrle 29541 . . . . . . . 8  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<_  Y )
2827ex 423 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  X  .<_  Y ) )
29283adant3r3 1162 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X C Y  ->  X  .<_  Y ) )
30293impia 1148 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<_  Y )
31 breq2 4029 . . . . 5  |-  ( Z  =  Y  ->  ( X  .<_  Z  <->  X  .<_  Y ) )
3230, 31syl5ibrcom 213 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  X  .<_  Z ) )
3326, 32jaod 369 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  X  .<_  Z ) )
34 breq1 4028 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Y  <->  Z  .<_  Y ) )
3530, 34syl5ibcom 211 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<_  Y ) )
36 breq2 4029 . . . . 5  |-  ( Z  =  Y  ->  ( Z  .<_  Z  <->  Z  .<_  Y ) )
3724, 36syl5ibcom 211 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  Z  .<_  Y ) )
3835, 37jaod 369 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  Z  .<_  Y ) )
3933, 38jcad 519 . 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 183 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257   Basecbs 13150   lecple 13217   Posetcpo 14076   ltcplt 14077    <o ccvr 29525
This theorem is referenced by:  cvrcmp  29546  leatb  29555  2llnmat  29786  2lnat  30046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-poset 14082  df-plt 14094  df-covers 29529
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