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Theorem cvnsym 23641
Description: The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnsym  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  -.  B  <oH  A ) )

Proof of Theorem cvnsym
StepHypRef Expression
1 cvpss 23636 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A  C.  B ) )
2 cvpss 23636 . . . . 5  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  <oH  A  ->  B  C.  A ) )
32ancoms 440 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( B  <oH  A  ->  B  C.  A ) )
4 pssn2lp 3391 . . . . 5  |-  -.  ( B  C.  A  /\  A  C.  B )
54imnani 413 . . . 4  |-  ( B 
C.  A  ->  -.  A  C.  B )
63, 5syl6 31 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( B  <oH  A  ->  -.  A  C.  B ) )
76con2d 109 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  B  ->  -.  B  <oH  A ) )
81, 7syld 42 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  -.  B  <oH  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1717    C. wpss 3264   class class class wbr 4153   CHcch 22280    <oH ccv 22315
This theorem is referenced by:  cvnref  23642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-cv 23630
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