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Theorem cvrnbtwn4 28599
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22794 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 2256 . . . 4  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . . 4  |-  C  =  (  <o  `  K )
41, 2, 3cvrnbtwn 28591 . . 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 415 . . . . 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 14021 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X ( lt `  K ) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
873adant3r2 1166 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X
( lt `  K
) Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
96, 2pltval 14021 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
1093com23 1162 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Z ( lt `  K ) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
11103adant3r1 1165 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Z
( lt `  K
) Y  <->  ( Z  .<_  Y  /\  Z  =/= 
Y ) ) )
128, 11anbi12d 694 . . . . . . 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 2504 . . . . . . . . 9  |-  ( ( X  =/=  Z  /\  Z  =/=  Y )  <->  -.  ( X  =  Z  \/  Z  =  Y )
)
1413anbi2i 678 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y
) ) )
15 an4 800 . . . . . . . 8  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  ( X  =/=  Z  /\  Z  =/=  Y
) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1614, 15bitr3i 244 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<_  Y )  /\  -.  ( X  =  Z  \/  Z  =  Y ) )  <->  ( ( X  .<_  Z  /\  X  =/=  Z )  /\  ( Z  .<_  Y  /\  Z  =/=  Y ) ) )
1712, 16syl6rbbr 257 . . . . . 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 287 . . . . 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 251 . . . 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 980 . . 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 203 . 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 14012 . . . . . . 7  |-  ( ( K  e.  Poset  /\  Z  e.  B )  ->  Z  .<_  Z )
23223ad2antr3 1127 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  Z  .<_  Z )
24233adant3 980 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Z  .<_  Z )
25 breq1 3966 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Z  <->  Z  .<_  Z ) )
2624, 25syl5ibrcom 215 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
271, 6, 3cvrle 28598 . . . . . . . 8  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<_  Y )
2827ex 425 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  X  .<_  Y ) )
29283adant3r3 1167 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X C Y  ->  X  .<_  Y ) )
30293impia 1153 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<_  Y )
31 breq2 3967 . . . . 5  |-  ( Z  =  Y  ->  ( X  .<_  Z  <->  X  .<_  Y ) )
3230, 31syl5ibrcom 215 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  X  .<_  Z ) )
3326, 32jaod 371 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  X  .<_  Z ) )
34 breq1 3966 . . . . 5  |-  ( X  =  Z  ->  ( X  .<_  Y  <->  Z  .<_  Y ) )
3530, 34syl5ibcom 213 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<_  Y ) )
36 breq2 3967 . . . . 5  |-  ( Z  =  Y  ->  ( Z  .<_  Z  <->  Z  .<_  Y ) )
3724, 36syl5ibcom 213 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( Z  =  Y  ->  Z  .<_  Y ) )
3835, 37jaod 371 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X  =  Z  \/  Z  =  Y )  ->  Z  .<_  Y ) )
3933, 38jcad 521 . 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 185 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638   Basecbs 13075   lecple 13142   Posetcpo 14001   ltcplt 14002    <o ccvr 28582
This theorem is referenced by:  cvrcmp  28603  leatb  28612  2llnmat  28843  2lnat  29103
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-poset 14007  df-plt 14019  df-covers 28586
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