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Metamath Proof Explorer

Theorem lcvnbtwn2

Description: The covers relation implies no in-betweenness. ( cvnbtwn2 analog.) (Contributed by NM, 7-Jan-2015)

Ref Expression
Hypotheses lcvnbtwn.s 
|- S = ( LSubSp ` W )
lcvnbtwn.c 
|- C = (
    lcvnbtwn.w 
    |- ( ph -> W e. X )
    lcvnbtwn.r 
    |- ( ph -> R e. S )
    lcvnbtwn.t 
    |- ( ph -> T e. S )
    lcvnbtwn.u 
    |- ( ph -> U e. S )
    lcvnbtwn.d 
    |- ( ph -> R C T )
    lcvnbtwn2.p 
    |- ( ph -> R C. U )
    lcvnbtwn2.q 
    |- ( ph -> U C_ T )
    Assertion lcvnbtwn2 
    |- ( ph -> U = T )

    Proof

    Step Hyp Ref Expression
    1 lcvnbtwn.s 
     |-  S = ( LSubSp ` W )
    2 lcvnbtwn.c 
     |-  C = (
      3 lcvnbtwn.w 
       |-  ( ph -> W e. X )
      4 lcvnbtwn.r 
       |-  ( ph -> R e. S )
      5 lcvnbtwn.t 
       |-  ( ph -> T e. S )
      6 lcvnbtwn.u 
       |-  ( ph -> U e. S )
      7 lcvnbtwn.d 
       |-  ( ph -> R C T )
      8 lcvnbtwn2.p 
       |-  ( ph -> R C. U )
      9 lcvnbtwn2.q 
       |-  ( ph -> U C_ T )
      10 1 2 3 4 5 6 7 lcvnbtwn 
       |-  ( ph -> -. ( R C. U /\ U C. T ) )
      11 iman 
       |-  ( ( ( R C. U /\ U C_ T ) -> U = T ) <-> -. ( ( R C. U /\ U C_ T ) /\ -. U = T ) )
      12 anass 
       |-  ( ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> ( R C. U /\ ( U C_ T /\ -. U = T ) ) )
      13 dfpss2 
       |-  ( U C. T <-> ( U C_ T /\ -. U = T ) )
      14 13 anbi2i 
       |-  ( ( R C. U /\ U C. T ) <-> ( R C. U /\ ( U C_ T /\ -. U = T ) ) )
      15 12 14 bitr4i 
       |-  ( ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> ( R C. U /\ U C. T ) )
      16 15 notbii 
       |-  ( -. ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> -. ( R C. U /\ U C. T ) )
      17 11 16 bitr2i 
       |-  ( -. ( R C. U /\ U C. T ) <-> ( ( R C. U /\ U C_ T ) -> U = T ) )
      18 10 17 sylib 
       |-  ( ph -> ( ( R C. U /\ U C_ T ) -> U = T ) )
      19 8 9 18 mp2and 
       |-  ( ph -> U = T )