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Theorem cvnbtwn 22866
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 22862 . . . 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 3264 . . . . . . . . 9  |-  ( x  =  C  ->  ( A  C.  x  <->  A  C.  C ) )
3 psseq1 3263 . . . . . . . . 9  |-  ( x  =  C  ->  (
x  C.  B  <->  C  C.  B ) )
42, 3anbi12d 691 . . . . . . . 8  |-  ( x  =  C  ->  (
( A  C.  x  /\  x  C.  B )  <-> 
( A  C.  C  /\  C  C.  B ) ) )
54rspcev 2884 . . . . . . 7  |-  ( ( C  e.  CH  /\  ( A  C.  C  /\  C  C.  B ) )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
65ex 423 . . . . . 6  |-  ( C  e.  CH  ->  (
( A  C.  C  /\  C  C.  B )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) )
76con3rr3 128 . . . . 5  |-  ( -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B )  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) )
87adantl 452 . . . 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 219 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) ) )
109com23 72 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  e.  CH  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) ) )
11103impia 1148 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C. wpss 3153   class class class wbr 4023   CHcch 21509    <oH ccv 21544
This theorem is referenced by:  cvnbtwn2  22867  cvnbtwn3  22868  cvnbtwn4  22869  cvntr  22872
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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