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Theorem lhpexle3 30201
Description: There exists atom under a co-atom different from any three 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
lhpexle3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
Distinct variable groups:    .<_ , p    A, p    H, p    K, p    W, p    X, p    Y, p    Z, p

Proof of Theorem lhpexle3
StepHypRef Expression
1 lhpex1.l . . . . 5  |-  .<_  =  ( le `  K )
2 lhpex1.a . . . . 5  |-  A  =  ( Atoms `  K )
3 lhpex1.h . . . . 5  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle2 30199 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
5 3anass 938 . . . . 5  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
65rexbii 2568 . . . 4  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
74, 6sylib 188 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
) ) )
81, 2, 3lhpexle2 30199 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z ) )
98adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Z ) )
10 3anass 938 . . . . . . 7  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
1110rexbii 2568 . . . . . 6  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
129, 11sylib 188 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
131, 2, 3lhpexle2 30199 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z ) )
14 3anass 938 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1514rexbii 2568 . . . . . . . . . 10  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1613, 15sylib 188 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
) ) )
17163ad2ant1 976 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
) ) )
18 simpl1 958 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
19 simpl3l 1010 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  Y  e.  A )
20 simpl2l 1008 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  Z  e.  A )
21 simprl 732 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  e.  A )
22 simpl3r 1011 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  Y  .<_  W )
23 simpl2r 1009 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  Z  .<_  W )
24 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  X  .<_  W )
251, 2, 3lhpexle3lem 30200 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  A  /\  Z  e.  A  /\  X  e.  A )  /\  ( Y  .<_  W  /\  Z  .<_  W  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/= 
X ) ) )
2618, 19, 20, 21, 22, 23, 24, 25syl133anc 1205 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) ) )
27 df-3an 936 . . . . . . . . . . . 12  |-  ( ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X )  <->  ( (
p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
2827anbi2i 675 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
29 3anass 938 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
3028, 29bitr4i 243 . . . . . . . . . 10  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
3130rexbii 2568 . . . . . . . . 9  |-  ( E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
3226, 31sylib 188 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X ) )
3317, 32lhpexle1lem 30196 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X ) )
34 an31 775 . . . . . . . . . 10  |-  ( ( ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
3534anbi2i 675 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
Y  /\  p  =/=  Z )  /\  p  =/= 
X ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
36 3anass 938 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
3735, 29, 363bitr4i 268 . . . . . . . 8  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )
3837rexbii 2568 . . . . . . 7  |-  ( E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y ) )
3933, 38sylib 188 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y ) )
40393expa 1151 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
4112, 40lhpexle1lem 30196 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
42 an32 773 . . . . . . 7  |-  ( ( ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
4342anbi2i 675 . . . . . 6  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
44 3anass 938 . . . . . 6  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
4543, 36, 443bitr4i 268 . . . . 5  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y )  /\  p  =/= 
Z ) )
4645rexbii 2568 . . . 4  |-  ( E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z ) )
4741, 46sylib 188 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
487, 47lhpexle1lem 30196 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z ) )
49 df-3an 936 . . . . 5  |-  ( ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( (
p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
5049anbi2i 675 . . . 4  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
5144, 50bitr4i 243 . . 3  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
5251rexbii 2568 . 2  |-  ( E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
5348, 52sylib 188 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdlemftr3  30754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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