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Theorem ltcvrntr 32698
Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
ltltncvr.b  |-  B  =  ( Base `  K
)
ltltncvr.s  |-  .<  =  ( lt `  K )
ltltncvr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
ltcvrntr  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y C Z )  ->  -.  X C Z ) )

Proof of Theorem ltcvrntr
StepHypRef Expression
1 ltltncvr.b . . . . 5  |-  B  =  ( Base `  K
)
2 ltltncvr.s . . . . 5  |-  .<  =  ( lt `  K )
3 ltltncvr.c . . . . 5  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 32545 . . . 4  |-  ( ( ( K  e.  A  /\  Y  e.  B  /\  Z  e.  B
)  /\  Y C Z )  ->  Y  .<  Z )
54ex 435 . . 3  |-  ( ( K  e.  A  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  ->  Y  .<  Z ) )
653adant3r1 1214 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y C Z  ->  Y  .<  Z ) )
71, 2, 3ltltncvr 32697 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )
86, 7sylan2d 484 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y C Z )  ->  -.  X C Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601   Basecbs 15084   ltcplt 16137    <o ccvr 32537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-covers 32541
This theorem is referenced by:  cvrntr  32699
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