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Theorem cvrnbtwn4 30091
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22885 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 2296 . . . 4  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . . 4  |-  C  =  (  <o  `  K )
41, 2, 3cvrnbtwn 30083 . . 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 14110 . . . . . . . . 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 14110 . . . . . . . . . 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 2544 . . . . . . . . 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 14101 . . . . . . 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 4042 . . . . 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 30090 . . . . . . . 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 4043 . . . . 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 4042 . . . . 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 4043 . . . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091    <o ccvr 30074
This theorem is referenced by:  cvrcmp  30095  leatb  30104  2llnmat  30335  2lnat  30595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-poset 14096  df-plt 14108  df-covers 30078
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