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Theorem ltltncvr 30121
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 731 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  K  e.  A )
2 simplr1 999 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  X  e.  B )
3 simplr3 1001 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Z  e.  B )
4 simplr2 1000 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Y  e.  B )
5 simpr 448 . . . 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 29970 . . . 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 1197 . . 3  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) )
1110ex 424 . 2  |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X C Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1211con2d 109 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13459   ltcplt 14388    <o ccvr 29961
This theorem is referenced by:  ltcvrntr  30122
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-covers 29965
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