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Theorem cvrntr 30236
Description: The covers relation is not transitive. (cvntr 22888 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 2296 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrntr.c . . . . 5  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 30082 . . . 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 30235 . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   ltcplt 14091    <o ccvr 30074
This theorem is referenced by:  atcvr0eq  30237  lnnat  30238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-covers 30078
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