HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cvnbtwn3 Unicode version

Theorem cvnbtwn3 22868
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 22866 . 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 )  ->  A  =  C )  <->  -.  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
3 eqcom 2285 . . . 4  |-  ( C  =  A  <->  A  =  C )
43imbi2i 303 . . 3  |-  ( ( ( A  C_  C  /\  C  C.  B )  ->  C  =  A )  <->  ( ( A 
C_  C  /\  C  C.  B )  ->  A  =  C ) )
5 dfpss2 3261 . . . . . 6  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
65anbi1i 676 . . . . 5  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  -.  A  =  C )  /\  C  C.  B ) )
7 an32 773 . . . . 5  |-  ( ( ( A  C_  C  /\  -.  A  =  C )  /\  C  C.  B )  <->  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
86, 7bitri 240 . . . 4  |-  ( ( A  C.  C  /\  C  C.  B )  <->  ( ( A  C_  C  /\  C  C.  B )  /\  -.  A  =  C )
)
98notbii 287 . . 3  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <->  -.  ( ( A  C_  C  /\  C  C.  B
)  /\  -.  A  =  C ) )
102, 4, 93bitr4ri 269 . 2  |-  ( -.  ( A  C.  C  /\  C  C.  B )  <-> 
( ( A  C_  C  /\  C  C.  B
)  ->  C  =  A ) )
111, 10syl6ib 217 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152    C. wpss 3153   class class class wbr 4023   CHcch 21509    <oH ccv 21544
This theorem is referenced by:  atcveq0  22928  atcvatlem  22965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cv 22859
  Copyright terms: Public domain W3C validator