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Theorem cvrnbtwn3 30148
Description: The covers relation implies no in-betweenness. (cvnbtwn3 23796 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
cvrletr.b  |-  B  =  ( Base `  K
)
cvrletr.l  |-  .<_  =  ( le `  K )
cvrletr.s  |-  .<  =  ( lt `  K )
cvrletr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnbtwn3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<  Y )  <->  X  =  Z
) )

Proof of Theorem cvrnbtwn3
StepHypRef Expression
1 cvrletr.b . . . 4  |-  B  =  ( Base `  K
)
2 cvrletr.s . . . 4  |-  .<  =  ( lt `  K )
3 cvrletr.c . . . 4  |-  C  =  (  <o  `  K )
41, 2, 3cvrnbtwn 30143 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )
5 cvrletr.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
65, 2pltval 14422 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
763adant3r2 1164 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/=  Z
) ) )
873adant3 978 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  .<  Z  <-> 
( X  .<_  Z  /\  X  =/=  Z ) ) )
98anbi1d 687 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<  Z  /\  Z  .<  Y )  <->  ( ( X 
.<_  Z  /\  X  =/= 
Z )  /\  Z  .<  Y ) ) )
109notbid 287 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( -.  ( X  .<  Z  /\  Z  .<  Y )  <->  -.  (
( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y ) ) )
11 an32 775 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z ) )
12 df-ne 2603 . . . . . . . 8  |-  ( X  =/=  Z  <->  -.  X  =  Z )
1312anbi2i 677 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1411, 13bitri 242 . . . . . 6  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1514notbii 289 . . . . 5  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
16 iman 415 . . . . 5  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  ->  X  =  Z )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z ) )
1715, 16bitr4i 245 . . . 4  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
1810, 17syl6bb 254 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( -.  ( X  .<  Z  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) ) )
194, 18mpbid 203 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
201, 5posref 14413 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
21 breq2 4219 . . . . . 6  |-  ( X  =  Z  ->  ( X  .<_  X  <->  X  .<_  Z ) )
2220, 21syl5ibcom 213 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Z  ->  X 
.<_  Z ) )
23223ad2antr1 1123 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  X  .<_  Z ) )
24233adant3 978 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
25 simp1 958 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  K  e.  Poset )
26 simp21 991 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  e.  B
)
27 simp22 992 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Y  e.  B
)
28 simp3 960 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X C Y )
291, 2, 3cvrlt 30142 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
3025, 26, 27, 28, 29syl31anc 1188 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<  Y )
31 breq1 4218 . . . 4  |-  ( X  =  Z  ->  ( X  .<  Y  <->  Z  .<  Y ) )
3230, 31syl5ibcom 213 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<  Y ) )
3324, 32jcad 521 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  ( X  .<_  Z  /\  Z  .<  Y ) ) )
3419, 33impbid 185 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<  Y )  <->  X  =  Z
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402   ltcplt 14403    <o ccvr 30134
This theorem is referenced by:  atcvreq0  30186  cvratlem  30292
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-poset 14408  df-plt 14420  df-covers 30138
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