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Theorem lhpexle1 34822
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 34819 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 tru 1383 . . . . . 6  |- T.
65jctr 542 . . . . 5  |-  ( p 
.<_  W  ->  ( p  .<_  W  /\ T.  ) )
76reximi 2932 . . . 4  |-  ( E. p  e.  A  p 
.<_  W  ->  E. p  e.  A  ( p  .<_  W  /\ T.  ) )
84, 7syl 16 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  ) )
9 simpll 753 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  K  e.  HL )
10 simprl 755 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  e.  A )
11 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3lhpbase 34812 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1312ad2antlr 726 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  W  e.  ( Base `  K
) )
14 eqid 2467 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
151, 14, 2, 3lhplt 34814 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X
( lt `  K
) W )
1611, 14, 22atlt 34253 . . . . 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 1231 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
18 simp3r 1025 . . . . . . . 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 1059 . . . . . . . . 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 997 . . . . . . . . 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 1060 . . . . . . . . 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 15448 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A  /\  W  e.  H )  ->  ( p ( lt
`  K ) W  ->  p  .<_  W ) )
2319, 20, 21, 22syl3anc 1228 . . . . . . . 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 1395 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  -> T.  )
26 simp3l 1024 . . . . . . 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 1176 . . . . . 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 1198 . . . . 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 2938 . . . 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 34821 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
) )
32 3simpb 994 . . 3  |-  ( ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  ( p  .<_  W  /\  p  =/= 
X ) )
3332reximi 2932 . 2  |-  ( E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X ) )
3431, 33syl 16 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 369    /\ w3a 973    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588   Basecbs 14490   lecple 14562   ltcplt 15428   Atomscatm 34078   HLchlt 34165   LHypclh 34798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-lhyp 34802
This theorem is referenced by:  lhpexle2lem  34823  lhpexle2  34824  lhpex2leN  34827
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