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Theorem lhpexle1 33548
Description: There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
Hypotheses
Ref Expression
lhpex1.l  |-  .<_  =  ( le `  K )
lhpex1.a  |-  A  =  ( Atoms `  K )
lhpex1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpexle1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X ) )
Distinct variable groups:    .<_ , p    A, p    H, p    K, p    W, p    X, p

Proof of Theorem lhpexle1
StepHypRef Expression
1 lhpex1.l . . . . 5  |-  .<_  =  ( le `  K )
2 lhpex1.a . . . . 5  |-  A  =  ( Atoms `  K )
3 lhpex1.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle 33545 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 tru 1442 . . . . . 6  |- T.
65jctr 545 . . . . 5  |-  ( p 
.<_  W  ->  ( p  .<_  W  /\ T.  ) )
76reximi 2895 . . . 4  |-  ( E. p  e.  A  p 
.<_  W  ->  E. p  e.  A  ( p  .<_  W  /\ T.  ) )
84, 7syl 17 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  ) )
9 simpll 759 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  K  e.  HL )
10 simprl 763 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  e.  A )
11 eqid 2423 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3lhpbase 33538 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1312ad2antlr 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  W  e.  ( Base `  K
) )
14 eqid 2423 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
151, 14, 2, 3lhplt 33540 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X
( lt `  K
) W )
1611, 14, 22atlt 32979 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  A  /\  W  e.  ( Base `  K ) )  /\  X ( lt `  K ) W )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
179, 10, 13, 15, 16syl31anc 1268 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
18 simp3r 1035 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  p ( lt `  K ) W )
19 simp1ll 1069 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  K  e.  HL )
20 simp2 1007 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  p  e.  A )
21 simp1lr 1070 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  W  e.  H )
221, 14pltle 16212 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A  /\  W  e.  H )  ->  ( p ( lt
`  K ) W  ->  p  .<_  W ) )
2319, 20, 21, 22syl3anc 1265 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  ( p
( lt `  K
) W  ->  p  .<_  W ) )
2418, 23mpd 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  p  .<_  W )
25 a1tru 1454 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  -> T.  )
26 simp3l 1034 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  p  =/=  X )
2724, 25, 263jca 1186 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  ( p  .<_  W  /\ T.  /\  p  =/=  X ) )
28273expia 1208 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A
)  ->  ( (
p  =/=  X  /\  p ( lt `  K ) W )  ->  ( p  .<_  W  /\ T.  /\  p  =/=  X ) ) )
2928reximdva 2902 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  ( E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
) ) )
3017, 29mpd 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X ) )
318, 30lhpexle1lem 33547 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
) )
32 3simpb 1004 . . 3  |-  ( ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  ( p  .<_  W  /\  p  =/= 
X ) )
3332reximi 2895 . 2  |-  ( E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X ) )
3431, 33syl 17 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438   T. wtru 1439    e. wcel 1873    =/= wne 2619   E.wrex 2777   class class class wbr 4429   ` cfv 5607   Basecbs 15126   lecple 15202   ltcplt 16191   Atomscatm 32804   HLchlt 32891   LHypclh 33524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-op 4011  df-uni 4226  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-id 4774  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-preset 16178  df-poset 16196  df-plt 16209  df-lub 16225  df-glb 16226  df-join 16227  df-meet 16228  df-p0 16290  df-p1 16291  df-lat 16297  df-clat 16359  df-oposet 32717  df-ol 32719  df-oml 32720  df-covers 32807  df-ats 32808  df-atl 32839  df-cvlat 32863  df-hlat 32892  df-lhyp 33528
This theorem is referenced by:  lhpexle2lem  33549  lhpexle2  33550  lhpex2leN  33553
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