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Theorem lhpexle3 34826
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 34824 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
5 3anass 977 . . . . 5  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
65rexbii 2965 . . . 4  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
74, 6sylib 196 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
) ) )
81, 2, 3lhpexle2 34824 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z ) )
98adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Z ) )
10 3anass 977 . . . . . . 7  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
1110rexbii 2965 . . . . . 6  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
129, 11sylib 196 . . . . 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 34824 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z ) )
14 3anass 977 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1514rexbii 2965 . . . . . . . . . 10  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1613, 15sylib 196 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
) ) )
17163ad2ant1 1017 . . . . . . . 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 999 . . . . . . . . . 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 1051 . . . . . . . . . 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 1049 . . . . . . . . . 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 755 . . . . . . . . . 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 1052 . . . . . . . . . 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 1050 . . . . . . . . . 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 756 . . . . . . . . . 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 34825 . . . . . . . . . 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 1251 . . . . . . . . 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 975 . . . . . . . . . . . 12  |-  ( ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X )  <->  ( (
p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
2827anbi2i 694 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
29 3anass 977 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
3028, 29bitr4i 252 . . . . . . . . . 10  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
3130rexbii 2965 . . . . . . . . 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 196 . . . . . . . 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 34821 . . . . . . 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 798 . . . . . . . . . 10  |-  ( ( ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
3534anbi2i 694 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
Y  /\  p  =/=  Z )  /\  p  =/= 
X ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
36 3anass 977 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
3735, 29, 363bitr4i 277 . . . . . . . 8  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )
3837rexbii 2965 . . . . . . 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 196 . . . . . 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 1196 . . . . 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 34821 . . . 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 796 . . . . . . 7  |-  ( ( ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
4342anbi2i 694 . . . . . 6  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
44 3anass 977 . . . . . 6  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
4543, 36, 443bitr4i 277 . . . . 5  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y )  /\  p  =/= 
Z ) )
4645rexbii 2965 . . . 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 196 . . 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 34821 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z ) )
49 df-3an 975 . . . . 5  |-  ( ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( (
p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
5049anbi2i 694 . . . 4  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
5144, 50bitr4i 252 . . 3  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
5251rexbii 2965 . 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 196 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
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 5588   lecple 14562   Atomscatm 34078   HLchlt 34165   LHypclh 34798
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-lhyp 34802
This theorem is referenced by:  cdlemftr3  35379
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