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Theorem llni 30002
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 961 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  B )
2 breq1 4183 . . . 4  |-  ( p  =  P  ->  (
p C X  <->  P C X ) )
32rspcev 3020 . . 3  |-  ( ( P  e.  A  /\  P C X )  ->  E. p  e.  A  p C X )
433ad2antl3 1121 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  E. p  e.  A  p C X )
5 simpl1 960 . . 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 30000 . . 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 889 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   class class class wbr 4180   ` cfv 5421   Basecbs 13432    <o ccvr 29757   Atomscatm 29758   LLinesclln 29985
This theorem is referenced by:  llnle  30012  atcvrlln  30014  lplncvrlvol  30110
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-llines 29992
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