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Theorem cvnbtwn3 23791
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C_  C  /\  C  C.  B )  ->  C  =  A ) ) )

Proof of Theorem cvnbtwn3
StepHypRef Expression
1 cvnbtwn 23789 . 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 )  ->  A  =  C )  <->  -.  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
3 eqcom 2438 . . . 4  |-  ( C  =  A  <->  A  =  C )
43imbi2i 304 . . 3  |-  ( ( ( A  C_  C  /\  C  C.  B )  ->  C  =  A )  <->  ( ( A 
C_  C  /\  C  C.  B )  ->  A  =  C ) )
5 dfpss2 3432 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
65anbi1i 677 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  C  C.  B ) )
7 an32 774 . . . . 5  |-  ( ( ( A  C_  C  /\  -.  A  =  C )  /\  C  C.  B )  <->  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
86, 7bitri 241 . . . 4  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
98notbii 288 . . 3  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <->  -.  ( ( A  C_  C  /\  C  C.  B
)  /\  -.  A  =  C ) )
102, 4, 93bitr4ri 270 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C_  C  /\  C  C.  B
)  ->  C  =  A ) )
111, 10syl6ib 218 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  (
( A  C_  C  /\  C  C.  B )  ->  C  =  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320    C. wpss 3321   class class class wbr 4212   CHcch 22432    <oH ccv 22467
This theorem is referenced by:  atcveq0  23851  atcvatlem  23888
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cv 23782
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