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Theorem cvrntr 29614
Description: The covers relation is not transitive. (cvntr 22872 analog.) (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
cvrntr.b  |-  B  =  ( Base `  K
)
cvrntr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrntr  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X C Y  /\  Y C Z )  ->  -.  X C Z ) )

Proof of Theorem cvrntr
StepHypRef Expression
1 cvrntr.b . . . . 5  |-  B  =  ( Base `  K
)
2 eqid 2283 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrntr.c . . . . 5  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 29460 . . . 4  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X
( lt `  K
) Y )
54ex 423 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  X ( lt
`  K ) Y ) )
653adant3r3 1162 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Y  ->  X
( lt `  K
) Y ) )
71, 2, 3ltcvrntr 29613 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X ( lt
`  K ) Y  /\  Y C Z )  ->  -.  X C Z ) )
86, 7syland 467 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X C Y  /\  Y C Z )  ->  -.  X C Z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   ltcplt 14075    <o ccvr 29452
This theorem is referenced by:  atcvr0eq  29615  lnnat  29616
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-covers 29456
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