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Theorem cvntr 25701
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 25694 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
213adant3 1008 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
3 cvpss 25694 . . 3  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
433adant1 1006 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
5 cvnbtwn 25695 . . . 4  |-  ( ( A  e.  CH  /\  C  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
653com23 1193 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
76con2d 115 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C.  B  /\  B  C.  C )  ->  -.  A  <oH  C ) )
82, 4, 7syl2and 483 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  <oH  B  /\  B  <oH  C )  ->  -.  A  <oH  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756    C. wpss 3334   class class class wbr 4297   CHcch 24336    <oH ccv 24371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-cv 25688
This theorem is referenced by:  atcv0eq  25788
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