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Theorem atbtwnexOLDN 34243
Description: There exists a 3rd atom  r on a line  P  .\/  Q intersecting element  X at  P, such that  r is different from  Q and not in  X. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atbtwn.b  |-  B  =  ( Base `  K
)
atbtwn.l  |-  .<_  =  ( le `  K )
atbtwn.j  |-  .\/  =  ( join `  K )
atbtwn.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atbtwnexOLDN  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    B, r    K, r    .<_ , r    P, r    Q, r    X, r
Allowed substitution hint:    .\/ ( r)

Proof of Theorem atbtwnexOLDN
StepHypRef Expression
1 simpr2 1003 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
2 simpr3 1004 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
3 nbrne2 4465 . . . 4  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
41, 2, 3syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
5 atbtwn.l . . . 4  |-  .<_  =  ( le `  K )
6 atbtwn.j . . . 4  |-  .\/  =  ( join `  K )
7 atbtwn.a . . . 4  |-  A  =  ( Atoms `  K )
85, 6, 7hlsupr 34182 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
94, 8syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
10 simp32 1033 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  Q )
11 simp31 1032 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  =/=  P )
12 simp1l 1020 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
13 simp2 997 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  e.  A )
14 simp1r1 1092 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  X  e.  B )
15 simp1r2 1093 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  X )
16 simp1r3 1094 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  Q  .<_  X )
17 simp33 1034 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  r  .<_  ( P  .\/  Q ) )
18 atbtwn.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1918, 5, 6, 7atbtwn 34242 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( r  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )  ->  (
r  =/=  P  <->  -.  r  .<_  X ) )
2012, 13, 14, 15, 16, 17, 19syl123anc 1245 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  P  <->  -.  r  .<_  X ) )
2111, 20mpbid 210 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  -.  r  .<_  X )
2210, 21, 173jca 1176 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  r  e.  A  /\  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) )  ->  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
23223exp 1195 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( r  e.  A  ->  ( ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  (
r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) ) ) )
2423reximdvai 2935 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  ( E. r  e.  A  (
r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P  .\/  Q
) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) ) )
259, 24mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   Atomscatm 34060   HLchlt 34147
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by: (None)
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