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Theorem cvrcmp 29525
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 29523 . . . . 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 29520 . . . . . . 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 29521 . . . . . . 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 2605 . . . . 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 14178 . . . 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 4106 . . 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 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4102   ` cfv 5334   Basecbs 13239   lecple 13306   Posetcpo 14167    <o ccvr 29504
This theorem is referenced by:  cvrcmp2  29526  atcmp  29553  llncmp  29763  lplncmp  29803  lvolcmp  29858  lhp2lt  30242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-poset 14173  df-plt 14185  df-covers 29508
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