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Theorem cvntr 23787
Description: The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvntr  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  <oH  B  /\  B  <oH  C )  ->  -.  A  <oH  C ) )

Proof of Theorem cvntr
StepHypRef Expression
1 cvpss 23780 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
213adant3 977 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
3 cvpss 23780 . . 3  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
433adant1 975 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
5 cvnbtwn 23781 . . . 4  |-  ( ( A  e.  CH  /\  C  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
653com23 1159 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
76con2d 109 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C.  B  /\  B  C.  C )  ->  -.  A  <oH  C ) )
82, 4, 7syl2and 470 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  <oH  B  /\  B  <oH  C )  ->  -.  A  <oH  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    C. wpss 3313   class class class wbr 4204   CHcch 22424    <oH ccv 22459
This theorem is referenced by:  atcv0eq  23874
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cv 23774
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