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Theorem cvnbtwn4 22794
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 22791 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
2 iman 415 . . 3  |-  ( ( ( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) )  <->  -.  (
( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) ) )
3 an4 800 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
4 ioran 478 . . . . . . 7  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  C  =  A  /\  -.  C  =  B ) )
5 eqcom 2258 . . . . . . . . 9  |-  ( C  =  A  <->  A  =  C )
65notbii 289 . . . . . . . 8  |-  ( -.  C  =  A  <->  -.  A  =  C )
76anbi1i 679 . . . . . . 7  |-  ( ( -.  C  =  A  /\  -.  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B
) )
84, 7bitri 242 . . . . . 6  |-  ( -.  ( C  =  A  \/  C  =  B )  <->  ( -.  A  =  C  /\  -.  C  =  B ) )
98anbi2i 678 . . . . 5  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( ( A  C_  C  /\  C  C_  B )  /\  ( -.  A  =  C  /\  -.  C  =  B ) ) )
10 dfpss2 3203 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
11 dfpss2 3203 . . . . . 6  |-  ( C 
C.  B  <->  ( C  C_  B  /\  -.  C  =  B ) )
1210, 11anbi12i 681 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  ( C  C_  B  /\  -.  C  =  B ) ) )
133, 9, 123bitr4i 270 . . . 4  |-  ( ( ( A  C_  C  /\  C  C_  B )  /\  -.  ( C  =  A  \/  C  =  B ) )  <->  ( A  C.  C  /\  C  C.  B ) )
1413notbii 289 . . 3  |-  ( -.  ( ( A  C_  C  /\  C  C_  B
)  /\  -.  ( C  =  A  \/  C  =  B )
)  <->  -.  ( A  C.  C  /\  C  C.  B ) )
152, 14bitr2i 243 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C_  C  /\  C  C_  B
)  ->  ( C  =  A  \/  C  =  B ) ) )
161, 15syl6ib 219 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 5    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3094    C. wpss 3095   class class class wbr 3963   CHcch 21434    <oH ccv 21469
This theorem is referenced by:  cvmdi  22829
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-cv 22784
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