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Theorem ltltncvr 29612
Description: A chained strong ordering is not a 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
ltltncvr  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )

Proof of Theorem ltltncvr
StepHypRef Expression
1 simpll 730 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  K  e.  A )
2 simplr1 997 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  X  e.  B )
3 simplr3 999 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Z  e.  B )
4 simplr2 998 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Y  e.  B )
5 simpr 447 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  X C Z )
6 ltltncvr.b . . . . 5  |-  B  =  ( Base `  K
)
7 ltltncvr.s . . . . 5  |-  .<  =  ( lt `  K )
8 ltltncvr.c . . . . 5  |-  C  =  (  <o  `  K )
96, 7, 8cvrnbtwn 29461 . . . 4  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Z  e.  B  /\  Y  e.  B
)  /\  X C Z )  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) )
101, 2, 3, 4, 5, 9syl131anc 1195 . . 3  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) )
1110ex 423 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1211con2d 107 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y  .<  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:  ltcvrntr  29613
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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