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Theorem lhpexlt 30191
Description: There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
lhpatltex.s  |-  .<  =  ( lt `  K )
lhpatltex.a  |-  A  =  ( Atoms `  K )
lhpatltex.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpexlt  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<  W )
Distinct variable groups:    A, p    K, p    .< , p    W, p
Allowed substitution hint:    H( p)

Proof of Theorem lhpexlt
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  HL )
2 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 lhpatltex.h . . . 4  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 30187 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54adantl 452 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
6 eqid 2283 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 eqid 2283 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
86, 7, 3lhp1cvr 30188 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
) ( 1. `  K ) )
9 lhpatltex.s . . 3  |-  .<  =  ( lt `  K )
10 lhpatltex.a . . 3  |-  A  =  ( Atoms `  K )
112, 9, 6, 7, 101cvratex 29662 . 2  |-  ( ( K  e.  HL  /\  W  e.  ( Base `  K )  /\  W
(  <o  `  K )
( 1. `  K
) )  ->  E. p  e.  A  p  .<  W )
121, 5, 8, 11syl3anc 1182 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   ltcplt 14075   1.cp1 14144    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  lhp0lt  30192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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