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Theorem llni 29756
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 960 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  B )
2 breq1 4128 . . . 4  |-  ( p  =  P  ->  (
p C X  <->  P C X ) )
32rspcev 2969 . . 3  |-  ( ( P  e.  A  /\  P C X )  ->  E. p  e.  A  p C X )
433ad2antl3 1120 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  E. p  e.  A  p C X )
5 simpl1 959 . . 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 29754 . . 3  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
115, 10syl 15 . 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 888 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   E.wrex 2629   class class class wbr 4125   ` cfv 5358   Basecbs 13356    <o ccvr 29511   Atomscatm 29512   LLinesclln 29739
This theorem is referenced by:  llnle  29766  atcvrlln  29768  lplncvrlvol  29864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-llines 29746
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