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Theorem lhpexle3 33315
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 33313 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
5 3anass 986 . . . . 5  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
65rexbii 2925 . . . 4  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
74, 6sylib 199 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
) ) )
81, 2, 3lhpexle2 33313 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z ) )
98adantr 466 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Z ) )
10 3anass 986 . . . . . . 7  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
1110rexbii 2925 . . . . . 6  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
129, 11sylib 199 . . . . 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 33313 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z ) )
14 3anass 986 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1514rexbii 2925 . . . . . . . . . 10  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1613, 15sylib 199 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
) ) )
17163ad2ant1 1026 . . . . . . . 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 1008 . . . . . . . . . 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 1060 . . . . . . . . . 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 1058 . . . . . . . . . 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 762 . . . . . . . . . 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 1061 . . . . . . . . . 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 1059 . . . . . . . . . 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 764 . . . . . . . . . 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 33314 . . . . . . . . . 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 1287 . . . . . . . . 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 984 . . . . . . . . . . . 12  |-  ( ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X )  <->  ( (
p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
2827anbi2i 698 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
29 3anass 986 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
3028, 29bitr4i 255 . . . . . . . . . 10  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
3130rexbii 2925 . . . . . . . . 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 199 . . . . . . . 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 33310 . . . . . . 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 807 . . . . . . . . . 10  |-  ( ( ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
3534anbi2i 698 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
Y  /\  p  =/=  Z )  /\  p  =/= 
X ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
36 3anass 986 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
3735, 29, 363bitr4i 280 . . . . . . . 8  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )
3837rexbii 2925 . . . . . . 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 199 . . . . . 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 1205 . . . . 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 33310 . . . 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 805 . . . . . . 7  |-  ( ( ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
4342anbi2i 698 . . . . . 6  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
44 3anass 986 . . . . . 6  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
4543, 36, 443bitr4i 280 . . . . 5  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y )  /\  p  =/= 
Z ) )
4645rexbii 2925 . . . 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 199 . . 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 33310 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z ) )
49 df-3an 984 . . . . 5  |-  ( ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( (
p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
5049anbi2i 698 . . . 4  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
5144, 50bitr4i 255 . . 3  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
5251rexbii 2925 . 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 199 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774   class class class wbr 4417   ` cfv 5592   lecple 15149   Atomscatm 32567   HLchlt 32654   LHypclh 33287
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-oposet 32480  df-ol 32482  df-oml 32483  df-covers 32570  df-ats 32571  df-atl 32602  df-cvlat 32626  df-hlat 32655  df-lhyp 33291
This theorem is referenced by:  cdlemftr3  33870
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