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Theorem cvrcmp 29473
Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
Hypotheses
Ref Expression
cvrcmp.b  |-  B  =  ( Base `  K
)
cvrcmp.l  |-  .<_  =  ( le `  K )
cvrcmp.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrcmp  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  .<_  Y  <->  X  =  Y ) )

Proof of Theorem cvrcmp
StepHypRef Expression
1 simpl1 958 . . . . 5  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  K  e.  Poset )
2 simpl23 1035 . . . . 5  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Z  e.  B )
3 simpl21 1033 . . . . 5  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  X  e.  B )
4 simpl3l 1010 . . . . 5  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Z C X )
5 cvrcmp.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cvrcmp.c . . . . . 6  |-  C  =  (  <o  `  K )
75, 6cvrne 29471 . . . . 5  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  =/=  X
)
81, 2, 3, 4, 7syl31anc 1185 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Z  =/=  X )
9 cvrcmp.l . . . . . . . 8  |-  .<_  =  ( le `  K )
105, 9, 6cvrle 29468 . . . . . . 7  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  .<_  X )
111, 2, 3, 4, 10syl31anc 1185 . . . . . 6  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Z  .<_  X )
12 simpr 447 . . . . . 6  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  X  .<_  Y )
13 simpl22 1034 . . . . . . 7  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Y  e.  B )
14 simpl3r 1011 . . . . . . 7  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  Z C Y )
155, 9, 6cvrnbtwn4 29469 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( Z  e.  B  /\  Y  e.  B  /\  X  e.  B )  /\  Z C Y )  ->  ( ( Z 
.<_  X  /\  X  .<_  Y )  <->  ( Z  =  X  \/  X  =  Y ) ) )
161, 2, 13, 3, 14, 15syl131anc 1195 . . . . . 6  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  -> 
( ( Z  .<_  X  /\  X  .<_  Y )  <-> 
( Z  =  X  \/  X  =  Y ) ) )
1711, 12, 16mpbi2and 887 . . . . 5  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  -> 
( Z  =  X  \/  X  =  Y ) )
18 neor 2530 . . . . 5  |-  ( ( Z  =  X  \/  X  =  Y )  <->  ( Z  =/=  X  ->  X  =  Y )
)
1917, 18sylib 188 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  -> 
( Z  =/=  X  ->  X  =  Y ) )
208, 19mpd 14 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  X  =  Y )
2120ex 423 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  .<_  Y  ->  X  =  Y ) )
22 simp1 955 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  K  e.  Poset )
23 simp21 988 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  e.  B )
245, 9posref 14085 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
2522, 23, 24syl2anc 642 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  .<_  X )
26 breq2 4027 . . 3  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
2725, 26syl5ibcom 211 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  =  Y  ->  X 
.<_  Y ) )
2821, 27impbid 183 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  .<_  Y  <->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074    <o ccvr 29452
This theorem is referenced by:  cvrcmp2  29474  atcmp  29501  llncmp  29711  lplncmp  29751  lvolcmp  29806  lhp2lt  30190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-poset 14080  df-plt 14092  df-covers 29456
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