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Theorem lhpex2leN 30202
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 30194 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 simprr 733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  p  .<_  W )
61, 2, 3lhpexle1 30197 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
76adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
85, 7jca 518 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  (
p  .<_  W  /\  E. q  e.  A  (
q  .<_  W  /\  q  =/=  p ) ) )
9 necom 2527 . . . . . . . . 9  |-  ( p  =/=  q  <->  q  =/=  p )
1093anbi3i 1144 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p
) )
11 3anass 938 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1210, 11bitri 240 . . . . . . 7  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1312rexbii 2568 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  E. q  e.  A  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
14 r19.42v 2694 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  ( q  .<_  W  /\  q  =/=  p ) )  <-> 
( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) ) )
1513, 14bitr2i 241 . . . . 5  |-  ( ( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )  <->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) )
168, 15sylib 188 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q
) )
1716exp32 588 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( p  e.  A  ->  ( p  .<_  W  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) ) ) )
1817reximdvai 2653 . 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 14 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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