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Theorem islln4 29622
Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b  |-  B  =  ( Base `  K
)
llnset.c  |-  C  =  (  <o  `  K )
llnset.a  |-  A  =  ( Atoms `  K )
llnset.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
islln4  |-  ( ( K  e.  D  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p C X ) )
Distinct variable groups:    A, p    K, p    X, p
Allowed substitution hints:    B( p)    C( p)    D( p)    N( p)

Proof of Theorem islln4
StepHypRef Expression
1 llnset.b . . 3  |-  B  =  ( Base `  K
)
2 llnset.c . . 3  |-  C  =  (  <o  `  K )
3 llnset.a . . 3  |-  A  =  ( Atoms `  K )
4 llnset.n . . 3  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln 29621 . 2  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
65baibd 876 1  |-  ( ( K  e.  D  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p C X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   class class class wbr 4154   ` cfv 5395   Basecbs 13397    <o ccvr 29378   Atomscatm 29379   LLinesclln 29606
This theorem is referenced by:  islln3  29625  llncmp  29637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-llines 29613
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