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Theorem cvnbtwn 27939
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )

Proof of Theorem cvnbtwn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cvbr 27935 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
2 psseq2 3521 . . . . . . . . 9  |-  ( x  =  C  ->  ( A  C.  x  <->  A  C.  C
) )
3 psseq1 3520 . . . . . . . . 9  |-  ( x  =  C  ->  (
x  C.  B  <->  C  C.  B
) )
42, 3anbi12d 717 . . . . . . . 8  |-  ( x  =  C  ->  (
( A  C.  x  /\  x  C.  B )  <-> 
( A  C.  C  /\  C  C.  B ) ) )
54rspcev 3150 . . . . . . 7  |-  ( ( C  e.  CH  /\  ( A  C.  C  /\  C  C.  B ) )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
65ex 436 . . . . . 6  |-  ( C  e.  CH  ->  (
( A  C.  C  /\  C  C.  B )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) )
76con3rr3 142 . . . . 5  |-  ( -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B )  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) )
87adantl 468 . . . 4  |-  ( ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )  ->  ( C  e. 
CH  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
91, 8syl6bi 232 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) ) )
109com23 81 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  e.  CH  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) ) )
11103impia 1205 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738    C. wpss 3405   class class class wbr 4402   CHcch 26582    <oH ccv 26617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-cv 27932
This theorem is referenced by:  cvnbtwn2  27940  cvnbtwn3  27941  cvnbtwn4  27942  cvntr  27945
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