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Theorem cvrnbtwn3 30074
Description: The covers relation implies no in-betweenness. (cvnbtwn3 23791 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 30069 . . 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 14417 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
763adant3r2 1163 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/=  Z
) ) )
873adant3 977 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  .<  Z  <-> 
( X  .<_  Z  /\  X  =/=  Z ) ) )
98anbi1d 686 . . . . 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 286 . . . 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 774 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z ) )
12 df-ne 2601 . . . . . . . 8  |-  ( X  =/=  Z  <->  -.  X  =  Z )
1312anbi2i 676 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1411, 13bitri 241 . . . . . 6  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1514notbii 288 . . . . 5  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
16 iman 414 . . . . 5  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  ->  X  =  Z )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z ) )
1715, 16bitr4i 244 . . . 4  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
1810, 17syl6bb 253 . . 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 202 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
201, 5posref 14408 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
21 breq2 4216 . . . . . 6  |-  ( X  =  Z  ->  ( X  .<_  X  <->  X  .<_  Z ) )
2220, 21syl5ibcom 212 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Z  ->  X 
.<_  Z ) )
23223ad2antr1 1122 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  X  .<_  Z ) )
24233adant3 977 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
25 simp1 957 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  K  e.  Poset )
26 simp21 990 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  e.  B
)
27 simp22 991 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Y  e.  B
)
28 simp3 959 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X C Y )
291, 2, 3cvrlt 30068 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
3025, 26, 27, 28, 29syl31anc 1187 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<  Y )
31 breq1 4215 . . . 4  |-  ( X  =  Z  ->  ( X  .<  Y  <->  Z  .<  Y ) )
3230, 31syl5ibcom 212 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<  Y ) )
3324, 32jcad 520 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  ( X  .<_  Z  /\  Z  .<  Y ) ) )
3419, 33impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   ltcplt 14398    <o ccvr 30060
This theorem is referenced by:  atcvreq0  30112  cvratlem  30218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-poset 14403  df-plt 14415  df-covers 30064
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