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Theorem cvrnbtwn3 30088
Description: The covers relation implies no in-betweenness. (cvnbtwn3 22884 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 30083 . . 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 14110 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/= 
Z ) ) )
763adant3r2 1161 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X  .<_  Z  /\  X  =/=  Z
) ) )
873adant3 975 . . . . . 6  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  .<  Z  <-> 
( X  .<_  Z  /\  X  =/=  Z ) ) )
98anbi1d 685 . . . . 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 285 . . . 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 773 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z ) )
12 df-ne 2461 . . . . . . . 8  |-  ( X  =/=  Z  <->  -.  X  =  Z )
1312anbi2i 675 . . . . . . 7  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  /\  X  =/=  Z )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1411, 13bitri 240 . . . . . 6  |-  ( ( ( X  .<_  Z  /\  X  =/=  Z )  /\  Z  .<  Y )  <->  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
1514notbii 287 . . . . 5  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z )
)
16 iman 413 . . . . 5  |-  ( ( ( X  .<_  Z  /\  Z  .<  Y )  ->  X  =  Z )  <->  -.  ( ( X  .<_  Z  /\  Z  .<  Y )  /\  -.  X  =  Z ) )
1715, 16bitr4i 243 . . . 4  |-  ( -.  ( ( X  .<_  Z  /\  X  =/=  Z
)  /\  Z  .<  Y )  <->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
1810, 17syl6bb 252 . . 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 201 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
.<_  Z  /\  Z  .<  Y )  ->  X  =  Z ) )
201, 5posref 14101 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
21 breq2 4043 . . . . . 6  |-  ( X  =  Z  ->  ( X  .<_  X  <->  X  .<_  Z ) )
2220, 21syl5ibcom 211 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  ( X  =  Z  ->  X 
.<_  Z ) )
23223ad2antr1 1120 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  X  .<_  Z ) )
24233adant3 975 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  X  .<_  Z ) )
25 simp1 955 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  K  e.  Poset )
26 simp21 988 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  e.  B
)
27 simp22 989 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  Y  e.  B
)
28 simp3 957 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X C Y )
291, 2, 3cvrlt 30082 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
3025, 26, 27, 28, 29syl31anc 1185 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  X  .<  Y )
31 breq1 4042 . . . 4  |-  ( X  =  Z  ->  ( X  .<  Y  <->  Z  .<  Y ) )
3230, 31syl5ibcom 211 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  Z  .<  Y ) )
3324, 32jcad 519 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( X  =  Z  ->  ( X  .<_  Z  /\  Z  .<  Y ) ) )
3419, 33impbid 183 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 176    /\ 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:  atcvreq0  30126  cvratlem  30232
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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