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

Theorem cvnbtwn 23789
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )

Proof of Theorem cvnbtwn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cvbr 23785 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A 
C.  x  /\  x  C.  B ) ) ) )
2 psseq2 3435 . . . . . . . . 9  |-  ( x  =  C  ->  ( A  C.  x  <->  A  C.  C ) )
3 psseq1 3434 . . . . . . . . 9  |-  ( x  =  C  ->  (
x  C.  B  <->  C  C.  B ) )
42, 3anbi12d 692 . . . . . . . 8  |-  ( x  =  C  ->  (
( A  C.  x  /\  x  C.  B )  <-> 
( A  C.  C  /\  C  C.  B ) ) )
54rspcev 3052 . . . . . . 7  |-  ( ( C  e.  CH  /\  ( A  C.  C  /\  C  C.  B ) )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )
65ex 424 . . . . . 6  |-  ( C  e.  CH  ->  (
( A  C.  C  /\  C  C.  B )  ->  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) )
76con3rr3 130 . . . . 5  |-  ( -. 
E. x  e.  CH  ( A  C.  x  /\  x  C.  B )  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) )
87adantl 453 . . . 4  |-  ( ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) )  ->  ( C  e. 
CH  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
91, 8syl6bi 220 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  -> 
( C  e.  CH  ->  -.  ( A  C.  C  /\  C  C.  B
) ) ) )
109com23 74 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( C  e.  CH  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) ) )
11103impia 1150 1  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    C. wpss 3321   class class class wbr 4212   CHcch 22432    <oH ccv 22467
This theorem is referenced by:  cvnbtwn2  23790  cvnbtwn3  23791  cvnbtwn4  23792  cvntr  23795
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
  Copyright terms: Public domain W3C validator