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Theorem llni 30379
Description: Condition implying a lattice line. (Contributed by NM, 17-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
llni  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  N )

Proof of Theorem llni
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  B )
2 breq1 4218 . . . 4  |-  ( p  =  P  ->  (
p C X  <->  P C X ) )
32rspcev 3054 . . 3  |-  ( ( P  e.  A  /\  P C X )  ->  E. p  e.  A  p C X )
433ad2antl3 1122 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  E. p  e.  A  p C X )
5 simpl1 961 . . 3  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  K  e.  D )
6 llnset.b . . . 4  |-  B  =  ( Base `  K
)
7 llnset.c . . . 4  |-  C  =  (  <o  `  K )
8 llnset.a . . . 4  |-  A  =  ( Atoms `  K )
9 llnset.n . . . 4  |-  N  =  ( LLines `  K )
106, 7, 8, 9islln 30377 . . 3  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
115, 10syl 16 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
121, 4, 11mpbir2and 890 1  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4215   ` cfv 5457   Basecbs 13474    <o ccvr 30134   Atomscatm 30135   LLinesclln 30362
This theorem is referenced by:  llnle  30389  atcvrlln  30391  lplncvrlvol  30487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-llines 30369
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