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Theorem lhpexle2 34806
Description: There exists atom under a co-atom different from any two other elements. (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
lhpexle2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
Distinct variable groups:    .<_ , p    A, p    H, p    K, p    W, p    X, p    Y, p

Proof of Theorem lhpexle2
StepHypRef Expression
1 lhpex1.l . . 3  |-  .<_  =  ( le `  K )
2 lhpex1.a . . 3  |-  A  =  ( Atoms `  K )
3 lhpex1.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle1 34804 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X ) )
51, 2, 3lhpexle1 34804 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y ) )
65adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
Y ) )
71, 2, 3lhpexle2lem 34805 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Y  .<_  W )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  X ) )
873expa 1196 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
Y  /\  p  =/=  X ) )
96, 8lhpexle1lem 34803 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
Y  /\  p  =/=  X ) )
10 3ancomb 982 . . . 4  |-  ( ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  X )  <->  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Y ) )
1110rexbii 2965 . . 3  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  X )  <->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Y ) )
129, 11sylib 196 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Y ) )
134, 12lhpexle1lem 34803 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586   lecple 14558   Atomscatm 34060   HLchlt 34147   LHypclh 34780
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784
This theorem is referenced by:  lhpexle3lem  34807  lhpexle3  34808  cdlemj3  35619
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