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Theorem lhpexle1 30819
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 30816 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 tru 1312 . . . . . 6  |-  T.
65jctr 526 . . . . 5  |-  ( p 
.<_  W  ->  ( p  .<_  W  /\  T.  )
)
76reximi 2663 . . . 4  |-  ( E. p  e.  A  p 
.<_  W  ->  E. p  e.  A  ( p  .<_  W  /\  T.  )
)
84, 7syl 15 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  T.  ) )
9 simpll 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  K  e.  HL )
10 simprl 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  e.  A )
11 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3lhpbase 30809 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1312ad2antlr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  W  e.  ( Base `  K
) )
14 eqid 2296 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
151, 14, 2, 3lhplt 30811 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X
( lt `  K
) W )
1611, 14, 22atlt 30250 . . . . 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 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
18 simp3r 984 . . . . . . . 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 1018 . . . . . . . . 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 956 . . . . . . . . 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 1019 . . . . . . . . 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 14111 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A  /\  W  e.  H )  ->  ( p ( lt
`  K ) W  ->  p  .<_  W ) )
2319, 20, 21, 22syl3anc 1182 . . . . . . . 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 14 . . . . . . 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 1321 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  ->  T.  )
26 simp3l 983 . . . . . . 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 1132 . . . . . 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 1153 . . . . 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 2668 . . . 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 14 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  T.  /\  p  =/=  X ) )
318, 30lhpexle1lem 30818 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  T.  /\  p  =/=  X
) )
32 3simpb 953 . . 3  |-  ( ( p  .<_  W  /\  T.  /\  p  =/=  X
)  ->  ( p  .<_  W  /\  p  =/= 
X ) )
3332reximi 2663 . 2  |-  ( E. p  e.  A  ( p  .<_  W  /\  T.  /\  p  =/=  X
)  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X ) )
3431, 33syl 15 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   ltcplt 14091   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpexle2lem  30820  lhpexle2  30821  lhpex2leN  30824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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