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Theorem lhpexle1 33649
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 33646 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 tru 1373 . . . . . 6  |- T.
65jctr 542 . . . . 5  |-  ( p 
.<_  W  ->  ( p  .<_  W  /\ T.  ) )
76reximi 2821 . . . 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 2441 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3lhpbase 33639 . . . . . 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 2441 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
151, 14, 2, 3lhplt 33641 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X
( lt `  K
) W )
1611, 14, 22atlt 33080 . . . . 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 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
18 simp3r 1017 . . . . . . . 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 1051 . . . . . . . . 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 989 . . . . . . . . 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 1052 . . . . . . . . 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 15129 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A  /\  W  e.  H )  ->  ( p ( lt
`  K ) W  ->  p  .<_  W ) )
2319, 20, 21, 22syl3anc 1218 . . . . . . . 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 1385 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  -> T.  )
26 simp3l 1016 . . . . . . 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 1168 . . . . . 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 1189 . . . . 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 2826 . . . 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 33648 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
) )
32 3simpb 986 . . 3  |-  ( ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  ( p  .<_  W  /\  p  =/= 
X ) )
3332reximi 2821 . 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 965    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2604   E.wrex 2714   class class class wbr 4290   ` cfv 5416   Basecbs 14172   lecple 14243   ltcplt 15109   Atomscatm 32905   HLchlt 32992   LHypclh 33625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-lhyp 33629
This theorem is referenced by:  lhpexle2lem  33650  lhpexle2  33651  lhpex2leN  33654
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