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Theorem ltcvrntr 30221
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 30068 . . . 4  |-  ( ( ( K  e.  A  /\  Y  e.  B  /\  Z  e.  B
)  /\  Y C Z )  ->  Y  .<  Z )
54ex 424 . . 3  |-  ( ( K  e.  A  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  ->  Y  .<  Z ) )
653adant3r1 1162 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y C Z  ->  Y  .<  Z ) )
71, 2, 3ltltncvr 30220 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )
86, 7sylan2d 469 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   ltcplt 14398    <o ccvr 30060
This theorem is referenced by:  cvrntr  30222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-covers 30064
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