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Theorem lhpex2leN 34827
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 34819 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
5 simprr 756 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  p  .<_  W )
61, 2, 3lhpexle1 34822 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
76adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )
85, 7jca 532 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  (
p  .<_  W  /\  E. q  e.  A  (
q  .<_  W  /\  q  =/=  p ) ) )
9 necom 2736 . . . . . . . . 9  |-  ( p  =/=  q  <->  q  =/=  p )
1093anbi3i 1189 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p
) )
11 3anass 977 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1210, 11bitri 249 . . . . . . 7  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
1312rexbii 2965 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  E. q  e.  A  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
14 r19.42v 3016 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  ( q  .<_  W  /\  q  =/=  p ) )  <-> 
( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) ) )
1513, 14bitr2i 250 . . . . 5  |-  ( ( p  .<_  W  /\  E. q  e.  A  ( q  .<_  W  /\  q  =/=  p ) )  <->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) )
168, 15sylib 196 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  p  .<_  W ) )  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q
) )
1716exp32 605 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( p  e.  A  ->  ( p  .<_  W  ->  E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q ) ) ) )
1817reximdvai 2935 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588   lecple 14562   Atomscatm 34078   HLchlt 34165   LHypclh 34798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-lhyp 34802
This theorem is referenced by: (None)
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