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Theorem ltltncvr 35290
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 753 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  K  e.  A )
2 simplr1 1038 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  X  e.  B )
3 simplr3 1040 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Z  e.  B )
4 simplr2 1039 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Y  e.  B )
5 simpr 461 . . . 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 35139 . . . 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 1241 . . 3  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) )
1110ex 434 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1211con2d 115 1  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594   Basecbs 14644   ltcplt 15697    <o ccvr 35130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-covers 35134
This theorem is referenced by:  ltcvrntr  35291
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