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Theorem cvnbtwn4 23745
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 23742 . 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  =  A  \/  C  =  B ) )  <->  -.  (
( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) ) )
3 an4 798 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
4 ioran 477 . . . . . . 7  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  C  =  A  /\  -.  C  =  B ) )
5 eqcom 2406 . . . . . . . . 9  |-  ( C  =  A  <->  A  =  C )
65notbii 288 . . . . . . . 8  |-  ( -.  C  =  A  <->  -.  A  =  C )
76anbi1i 677 . . . . . . 7  |-  ( ( -.  C  =  A  /\  -.  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B
) )
84, 7bitri 241 . . . . . 6  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B ) )
98anbi2i 676 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) ) )
10 dfpss2 3392 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
11 dfpss2 3392 . . . . . 6  |-  ( C 
C.  B  <->  ( C  C_  B  /\  -.  C  =  B ) )
1210, 11anbi12i 679 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
133, 9, 123bitr4i 269 . . . 4  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( A  C.  C  /\  C  C.  B ) )
1413notbii 288 . . 3  |-  ( -.  ( ( A  C_  C  /\  C  C_  B
)  /\  -.  ( C  =  A  \/  C  =  B )
)  <->  -.  ( A  C.  C  /\  C  C.  B ) )
152, 14bitr2i 242 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C_  C  /\  C  C_  B
)  ->  ( C  =  A  \/  C  =  B ) ) )
161, 15syl6ib 218 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 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280    C. wpss 3281   class class class wbr 4172   CHcch 22385    <oH ccv 22420
This theorem is referenced by:  cvmdi  23780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cv 23735
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