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Theorem cvrcmp 32920
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 1033 . . . . 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 1110 . . . . 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 1108 . . . . 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 1085 . . . . 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 32918 . . . . 5  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  =/=  X
)
81, 2, 3, 4, 7syl31anc 1295 . . . 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 32915 . . . . . . 7  |-  ( ( ( K  e.  Poset  /\  Z  e.  B  /\  X  e.  B )  /\  Z C X )  ->  Z  .<_  X )
111, 2, 3, 4, 10syl31anc 1295 . . . . . 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 468 . . . . . 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 1109 . . . . . . 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 1086 . . . . . . 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 32916 . . . . . . 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 1305 . . . . . 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 935 . . . . 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 2734 . . . . 5  |-  ( ( Z  =  X  \/  X  =  Y )  <->  ( Z  =/=  X  ->  X  =  Y )
)
1917, 18sylib 201 . . . 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 441 . 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 1030 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  K  e.  Poset )
23 simp21 1063 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  e.  B )
245, 9posref 16274 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
2522, 23, 24syl2anc 673 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  X  .<_  X )
26 breq2 4399 . . 3  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
2725, 26syl5ibcom 228 . 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 195 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263    <o ccvr 32899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-preset 16251  df-poset 16269  df-plt 16282  df-covers 32903
This theorem is referenced by:  cvrcmp2  32921  atcmp  32948  llncmp  33158  lplncmp  33198  lvolcmp  33253  lhp2lt  33637
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