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Theorem islln4 30241
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 30240 . 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 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204   ` cfv 5446   Basecbs 13461    <o ccvr 29997   Atomscatm 29998   LLinesclln 30225
This theorem is referenced by:  islln3  30244  llncmp  30256
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-llines 30232
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