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Theorem cvrcmp 30155
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 961 . . . . 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 1038 . . . . 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 1036 . . . . 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 1013 . . . . 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 30153 . . . . 5  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  =/=  X
)
81, 2, 3, 4, 7syl31anc 1188 . . . 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 30150 . . . . . . 7  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  .<_  X )
111, 2, 3, 4, 10syl31anc 1188 . . . . . 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 449 . . . . . 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 1037 . . . . . . 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 1014 . . . . . . 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 30151 . . . . . . 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 1198 . . . . . 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 889 . . . . 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 2690 . . . . 5  |-  ( ( Z  =  X  \/  X  =  Y )  <->  ( Z  =/=  X  ->  X  =  Y )
)
1917, 18sylib 190 . . . 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 15 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  /\  X  .<_  Y )  ->  X  =  Y )
2120ex 425 . 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 958 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  K  e.  Poset )
23 simp21 991 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  e.  B )
245, 9posref 14413 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
2522, 23, 24syl2anc 644 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  .<_  X )
26 breq2 4219 . . 3  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
2725, 26syl5ibcom 213 . 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 185 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402    <o ccvr 30134
This theorem is referenced by:  cvrcmp2  30156  atcmp  30183  llncmp  30393  lplncmp  30433  lvolcmp  30488  lhp2lt  30872
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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