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Theorem cvnbtwn4 22830
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 22827 . 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 2260 . . . . . . . . 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 3236 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
11 dfpss2 3236 . . . . . 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 3127    C. wpss 3128   class class class wbr 3997   CHcch 21470    <oH ccv 21505
This theorem is referenced by:  cvmdi  22865
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-cv 22820
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