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Theorem cvnbtwn2 27939
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C.  C  /\  C  C_  B )  ->  C  =  B ) ) )

Proof of Theorem cvnbtwn2
StepHypRef Expression
1 cvnbtwn 27938 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
2 iman 425 . . 3  |-  ( ( ( A  C.  C  /\  C  C_  B )  ->  C  =  B )  <->  -.  ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )
)
3 anass 653 . . . . 5  |-  ( ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )  <->  ( A  C.  C  /\  ( C 
C_  B  /\  -.  C  =  B )
) )
4 dfpss2 3550 . . . . . 6  |-  ( C 
C.  B  <->  ( C  C_  B  /\  -.  C  =  B ) )
54anbi2i 698 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( A  C.  C  /\  ( C 
C_  B  /\  -.  C  =  B )
) )
63, 5bitr4i 255 . . . 4  |-  ( ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )  <->  ( A  C.  C  /\  C  C.  B ) )
76notbii 297 . . 3  |-  ( -.  ( ( A  C.  C  /\  C  C_  B
)  /\  -.  C  =  B )  <->  -.  ( A  C.  C  /\  C  C.  B ) )
82, 7bitr2i 253 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C.  C  /\  C  C_  B
)  ->  C  =  B ) )
91, 8syl6ib 229 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C.  C  /\  C  C_  B )  ->  C  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    C_ wss 3436    C. wpss 3437   class class class wbr 4423   CHcch 26581    <oH ccv 26616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-cv 27931
This theorem is referenced by:  cvati  28018  cvexchlem  28020  atexch  28033
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