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Theorem lhpex2leN 30261
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 30253 . 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 30256 . . . . . . 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 2610 . . . . . . . . 9  |-  ( p  =/=  q  <->  q  =/=  p )
1093anbi3i 1145 . . . . . . . 8  |-  ( ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  ( p  .<_  W  /\  q  .<_  W  /\  q  =/=  p
) )
11 3anass 939 . . . . . . . 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 2653 . . . . . 6  |-  ( E. q  e.  A  ( p  .<_  W  /\  q  .<_  W  /\  p  =/=  q )  <->  E. q  e.  A  ( p  .<_  W  /\  ( q 
.<_  W  /\  q  =/=  p ) ) )
14 r19.42v 2779 . . . . . 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 2738 . 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   class class class wbr 4125   ` cfv 5358   lecple 13423   Atomscatm 29512   HLchlt 29599   LHypclh 30232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-lhyp 30236
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