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Theorem lhpex2leN 30495
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 30487 . 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 30490 . . . . . . 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 2648 . . . . . . . . 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 2691 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  E. q  e.  A  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
14 r19.42v 2822 . . . . . 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 2776 . 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413   lecple 13491   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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