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Theorem lhpex2leN 30810
Description: There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhp2at.l  |-  .<_  =  ( le `  K )
lhp2at.a  |-  A  =  ( Atoms `  K )
lhp2at.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpex2leN  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) )
Distinct variable groups:    q, p, A    H, p, q    K, p, q    .<_ , p, q    W, p, q

Proof of Theorem lhpex2leN
StepHypRef Expression
1 lhp2at.l . . 3  |-  .<_  =  ( le `  K )
2 lhp2at.a . . 3  |-  A  =  ( Atoms `  K )
3 lhp2at.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexle 30802 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 simprr 734 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  p  .<_  W )
61, 2, 3lhpexle1 30805 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
76adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
85, 7jca 519 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  (
p  .<_  W  /\  E. q  e.  A  (
q  .<_  W  /\  q  =/=  p ) ) )
9 necom 2685 . . . . . . . . 9  |-  ( p  =/=  q  <->  q  =/=  p )
1093anbi3i 1146 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p
) )
11 3anass 940 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1210, 11bitri 241 . . . . . . 7  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1312rexbii 2730 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  E. q  e.  A  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
14 r19.42v 2862 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  ( q  .<_  W  /\  q  =/=  p ) )  <-> 
( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) ) )
1513, 14bitr2i 242 . . . . 5  |-  ( ( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )  <->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) )
168, 15sylib 189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q
) )
1716exp32 589 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( p  e.  A  ->  ( p  .<_  W  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) ) ) )
1817reximdvai 2816 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. p  e.  A  p  .<_  W  ->  E. p  e.  A  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) ) )
194, 18mpd 15 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) )
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 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
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