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Theorem ltltncvr 33386
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 1030 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  X  e.  B )
3 simplr3 1032 . . . 4  |-  ( ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Z )  ->  Z  e.  B )
4 simplr2 1031 . . . 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 33235 . . . 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 1232 . . 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521   Basecbs 14287   ltcplt 15225    <o ccvr 33226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-covers 33230
This theorem is referenced by:  ltcvrntr  33387
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