Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpexle1 Structured version   Unicode version

Theorem lhpexle1 33282
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 33279 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 tru 1441 . . . . . 6  |- T.
65jctr 544 . . . . 5  |-  ( p 
.<_  W  ->  ( p  .<_  W  /\ T.  ) )
76reximi 2900 . . . 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 758 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  K  e.  HL )
10 simprl 762 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  e.  A )
11 eqid 2429 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3lhpbase 33272 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1312ad2antlr 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  W  e.  ( Base `  K
) )
14 eqid 2429 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
151, 14, 2, 3lhplt 33274 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X
( lt `  K
) W )
1611, 14, 22atlt 32713 . . . . 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 1267 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  =/=  X  /\  p ( lt `  K ) W ) )
18 simp3r 1034 . . . . . . . 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 1068 . . . . . . . . 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 1006 . . . . . . . . 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 1069 . . . . . . . . 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 16158 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A  /\  W  e.  H )  ->  ( p ( lt
`  K ) W  ->  p  .<_  W ) )
2319, 20, 21, 22syl3anc 1264 . . . . . . . 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 1453 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W ) )  /\  p  e.  A  /\  ( p  =/=  X  /\  p ( lt `  K ) W ) )  -> T.  )
26 simp3l 1033 . . . . . . 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 1185 . . . . . 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 1207 . . . . 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 2907 . . . 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 33281 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\ T.  /\  p  =/=  X
) )
32 3simpb 1003 . . 3  |-  ( ( p  .<_  W  /\ T.  /\  p  =/=  X
)  ->  ( p  .<_  W  /\  p  =/= 
X ) )
3332reximi 2900 . 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 370    /\ w3a 982    = wceq 1437   T. wtru 1438    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601   Basecbs 15084   lecple 15159   ltcplt 16137   Atomscatm 32538   HLchlt 32625   LHypclh 33258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lhyp 33262
This theorem is referenced by:  lhpexle2lem  33283  lhpexle2  33284  lhpex2leN  33287
  Copyright terms: Public domain W3C validator