Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpexle3 Structured version   Unicode version

Theorem lhpexle3 30809
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 30807 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
5 3anass 940 . . . . 5  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
65rexbii 2730 . . . 4  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y ) ) )
74, 6sylib 189 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
) ) )
81, 2, 3lhpexle2 30807 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z ) )
98adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Z  e.  A  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Z ) )
10 3anass 940 . . . . . . 7  |-  ( ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
1110rexbii 2730 . . . . . 6  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z ) ) )
129, 11sylib 189 . . . . 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 30807 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z ) )
14 3anass 940 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1514rexbii 2730 . . . . . . . . . 10  |-  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  Y  /\  p  =/=  Z )  <->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z ) ) )
1613, 15sylib 189 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
) ) )
17163ad2ant1 978 . . . . . . . 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 960 . . . . . . . . . 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 1012 . . . . . . . . . 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 1010 . . . . . . . . . 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 733 . . . . . . . . . 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 1013 . . . . . . . . . 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 1011 . . . . . . . . . 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 734 . . . . . . . . . 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 30808 . . . . . . . . . 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 1207 . . . . . . . . 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 938 . . . . . . . . . . . 12  |-  ( ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X )  <->  ( (
p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
2827anbi2i 676 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
29 3anass 940 . . . . . . . . . . 11  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) ) )
3028, 29bitr4i 244 . . . . . . . . . 10  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z  /\  p  =/=  X
) )  <->  ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z )  /\  p  =/=  X ) )
3130rexbii 2730 . . . . . . . . 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 189 . . . . . . . 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 30804 . . . . . . 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 776 . . . . . . . . . 10  |-  ( ( ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) )
3534anbi2i 676 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
Y  /\  p  =/=  Z )  /\  p  =/= 
X ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
36 3anass 940 . . . . . . . . 9  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Z )  /\  p  =/=  Y ) ) )
3735, 29, 363bitr4i 269 . . . . . . . 8  |-  ( ( p  .<_  W  /\  ( p  =/=  Y  /\  p  =/=  Z
)  /\  p  =/=  X )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )
3837rexbii 2730 . . . . . . 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 189 . . . . . 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 1153 . . . . 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 30804 . . . 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 774 . . . . . . 7  |-  ( ( ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
4342anbi2i 676 . . . . . 6  |-  ( ( p  .<_  W  /\  ( ( p  =/= 
X  /\  p  =/=  Z )  /\  p  =/= 
Y ) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
44 3anass 940 . . . . . 6  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
4543, 36, 443bitr4i 269 . . . . 5  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Z
)  /\  p  =/=  Y )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y )  /\  p  =/= 
Z ) )
4645rexbii 2730 . . . 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 189 . . 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 30804 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z ) )
49 df-3an 938 . . . . 5  |-  ( ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z )  <->  ( (
p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) )
5049anbi2i 676 . . . 4  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z
) )  <->  ( p  .<_  W  /\  ( ( p  =/=  X  /\  p  =/=  Y )  /\  p  =/=  Z ) ) )
5144, 50bitr4i 244 . . 3  |-  ( ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y
)  /\  p  =/=  Z )  <->  ( p  .<_  W  /\  ( p  =/= 
X  /\  p  =/=  Y  /\  p  =/=  Z
) ) )
5251rexbii 2730 . 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 189 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454   lecple 13536   Atomscatm 30061   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdlemftr3  31362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
  Copyright terms: Public domain W3C validator