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Theorem cvnbtwn4 22885
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn4  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) ) ) )

Proof of Theorem cvnbtwn4
StepHypRef Expression
1 cvnbtwn 22882 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
2 iman 413 . . 3  |-  ( ( ( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) )  <->  -.  (
( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) ) )
3 an4 797 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
4 ioran 476 . . . . . . 7  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  C  =  A  /\  -.  C  =  B ) )
5 eqcom 2298 . . . . . . . . 9  |-  ( C  =  A  <->  A  =  C )
65notbii 287 . . . . . . . 8  |-  ( -.  C  =  A  <->  -.  A  =  C )
76anbi1i 676 . . . . . . 7  |-  ( ( -.  C  =  A  /\  -.  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B
) )
84, 7bitri 240 . . . . . 6  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B ) )
98anbi2i 675 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) ) )
10 dfpss2 3274 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
11 dfpss2 3274 . . . . . 6  |-  ( C 
C.  B  <->  ( C  C_  B  /\  -.  C  =  B ) )
1210, 11anbi12i 678 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
133, 9, 123bitr4i 268 . . . 4  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( A  C.  C  /\  C  C.  B ) )
1413notbii 287 . . 3  |-  ( -.  ( ( A  C_  C  /\  C  C_  B
)  /\  -.  ( C  =  A  \/  C  =  B )
)  <->  -.  ( A  C.  C  /\  C  C.  B ) )
152, 14bitr2i 241 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C_  C  /\  C  C_  B
)  ->  ( C  =  A  \/  C  =  B ) ) )
161, 15syl6ib 217 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165    C. wpss 3166   class class class wbr 4039   CHcch 21525    <oH ccv 21560
This theorem is referenced by:  cvmdi  22920
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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