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Theorem islln 30303
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
islln  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  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 islln
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 llnset.b . . . 4  |-  B  =  ( Base `  K
)
2 llnset.c . . . 4  |-  C  =  (  <o  `  K )
3 llnset.a . . . 4  |-  A  =  ( Atoms `  K )
4 llnset.n . . . 4  |-  N  =  ( LLines `  K )
51, 2, 3, 4llnset 30302 . . 3  |-  ( K  e.  D  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
65eleq2d 2503 . 2  |-  ( K  e.  D  ->  ( X  e.  N  <->  X  e.  { x  e.  B  |  E. p  e.  A  p C x } ) )
7 breq2 4216 . . . 4  |-  ( x  =  X  ->  (
p C x  <->  p C X ) )
87rexbidv 2726 . . 3  |-  ( x  =  X  ->  ( E. p  e.  A  p C x  <->  E. p  e.  A  p C X ) )
98elrab 3092 . 2  |-  ( X  e.  { x  e.  B  |  E. p  e.  A  p C x }  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) )
106, 9syl6bb 253 1  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   class class class wbr 4212   ` cfv 5454   Basecbs 13469    <o ccvr 30060   Atomscatm 30061   LLinesclln 30288
This theorem is referenced by:  islln4  30304  llni  30305  llnbase  30306  llnnleat  30310
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-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-llines 30295
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