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Theorem cvnbtwn2 23778
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 23777 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
2 iman 414 . . 3  |-  ( ( ( A  C.  C  /\  C  C_  B )  ->  C  =  B )  <->  -.  ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )
)
3 anass 631 . . . . 5  |-  ( ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )  <->  ( A  C.  C  /\  ( C  C_  B  /\  -.  C  =  B )
) )
4 dfpss2 3424 . . . . . 6  |-  ( C 
C.  B  <->  ( C  C_  B  /\  -.  C  =  B ) )
54anbi2i 676 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( A  C.  C  /\  ( C  C_  B  /\  -.  C  =  B )
) )
63, 5bitr4i 244 . . . 4  |-  ( ( ( A  C.  C  /\  C  C_  B )  /\  -.  C  =  B )  <->  ( A  C.  C  /\  C  C.  B ) )
76notbii 288 . . 3  |-  ( -.  ( ( A  C.  C  /\  C  C_  B
)  /\  -.  C  =  B )  <->  -.  ( A  C.  C  /\  C  C.  B ) )
82, 7bitr2i 242 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C.  C  /\  C  C_  B
)  ->  C  =  B ) )
91, 8syl6ib 218 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312    C. wpss 3313   class class class wbr 4204   CHcch 22420    <oH ccv 22455
This theorem is referenced by:  cvati  23857  cvexchlem  23859  atexch  23872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cv 23770
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