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Theorem cvntr 22872
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 22865 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
213adant3 975 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
3 cvpss 22865 . . 3  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
433adant1 973 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
5 cvnbtwn 22866 . . . 4  |-  ( ( A  e.  CH  /\  C  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
653com23 1157 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
76con2d 107 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C.  B  /\  B  C.  C )  ->  -.  A  <oH  C ) )
82, 4, 7syl2and 469 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 358    /\ w3a 934    e. wcel 1684    C. wpss 3153   class class class wbr 4023   CHcch 21509    <oH ccv 21544
This theorem is referenced by:  atcv0eq  22959
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cv 22859
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