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Theorem cvnbtwn 22882
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 22878 . . . 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 3277 . . . . . . . . 9  |-  ( x  =  C  ->  ( A  C.  x  <->  A  C.  C ) )
3 psseq1 3276 . . . . . . . . 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 2897 . . . . . . 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 1632    e. wcel 1696   E.wrex 2557    C. wpss 3166   class class class wbr 4039   CHcch 21525    <oH ccv 21560
This theorem is referenced by:  cvnbtwn2  22883  cvnbtwn3  22884  cvnbtwn4  22885  cvntr  22888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cv 22875
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