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Theorem cvntr 27921
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 27914 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
213adant3 1025 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
3 cvpss 27914 . . 3  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
433adant1 1023 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  <oH  C  ->  B  C.  C ) )
5 cvnbtwn 27915 . . . 4  |-  ( ( A  e.  CH  /\  C  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
653com23 1211 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  C  ->  -.  ( A  C.  B  /\  B  C.  C ) ) )
76con2d 118 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C.  B  /\  B  C.  C )  ->  -.  A  <oH  C ) )
82, 4, 7syl2and 485 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 370    /\ w3a 982    e. wcel 1867    C. wpss 3434   class class class wbr 4417   CHcch 26558    <oH ccv 26593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-cv 27908
This theorem is referenced by:  atcv0eq  28008
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