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Theorem lhpexle3 35298
Description: There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
Hypotheses
Ref Expression
lhpex1.l |- .<_ = ( le ` K )
lhpex1.a |- A = ( Atoms ` K )
lhpex1.h |- H = ( LHyp ` K )
Assertion
Ref Expression
lhpexle3 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
Distinct variable groups:   .<_ , p   A, p   H, p   K, p   W, p   X, p   Y, p   Z, p
Proof of Theorem lhpexle3
StepHypRef Expression
1 lhpex1.l . . . . 5 |- .<_ = ( le ` K )
2 lhpex1.a . . . . 5 |- A = ( Atoms ` K )
3 lhpex1.h . . . . 5 |- H = ( LHyp ` K )
41, 2, 3lhpexle2 35296 . . . 4 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) )
5 3anass 1042 . . . . 5 |- ( ( p .<_ W /\ p =/= X /\ p =/= Y ) <-> ( p .<_ W /\ ( p =/= X /\ p =/= Y ) ) )
65rexbii 3041 . . . 4 |- ( E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) <-> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) ) )
74, 6sylib 208 . . 3 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) ) )
81, 2, 3lhpexle2 35296 . . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) )
98adantr 481 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) )
10 3anass 1042 . . . . . . 7 |- ( ( p .<_ W /\ p =/= X /\ p =/= Z ) <-> ( p .<_ W /\ ( p =/= X /\ p =/= Z ) ) )
1110rexbii 3041 . . . . . 6 |- ( E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) <-> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) ) )
129, 11sylib 208 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) ) )
131, 2, 3lhpexle2 35296 . . . . . . . . . 10 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= Y /\ p =/= Z ) )
14 3anass 1042 . . . . . . . . . . 11 |- ( ( p .<_ W /\ p =/= Y /\ p =/= Z ) <-> ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) ) )
1514rexbii 3041 . . . . . . . . . 10 |- ( E. p e. A ( p .<_ W /\ p =/= Y /\ p =/= Z ) <-> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) ) )
1613, 15sylib 208 . . . . . . . . 9 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) ) )
17163ad2ant1 1082 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) ) )
18 simpl1 1064 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> ( K e. HL /\ W e. H ) )
19 simpl3l 1116 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> Y e. A )
20 simpl2l 1114 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> Z e. A )
21 simprl 794 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> X e. A )
22 simpl3r 1117 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> Y .<_ W )
23 simpl2r 1115 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> Z .<_ W )
24 simprr 796 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> X .<_ W )
251, 2, 3lhpexle3lem 35297 . . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( Y e. A /\ Z e. A /\ X e. A ) /\ ( Y .<_ W /\ Z .<_ W /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z /\ p =/= X ) ) )
2618, 19, 20, 21, 22, 23, 24, 25syl133anc 1349 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z /\ p =/= X ) ) )
27 df-3an 1039 . . . . . . . . . . . 12 |- ( ( p =/= Y /\ p =/= Z /\ p =/= X ) <-> ( ( p =/= Y /\ p =/= Z ) /\ p =/= X ) )
2827anbi2i 730 . . . . . . . . . . 11 |- ( ( p .<_ W /\ ( p =/= Y /\ p =/= Z /\ p =/= X ) ) <-> ( p .<_ W /\ ( ( p =/= Y /\ p =/= Z ) /\ p =/= X ) ) )
29 3anass 1042 . . . . . . . . . . 11 |- ( ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) <-> ( p .<_ W /\ ( ( p =/= Y /\ p =/= Z ) /\ p =/= X ) ) )
3028, 29bitr4i 267 . . . . . . . . . 10 |- ( ( p .<_ W /\ ( p =/= Y /\ p =/= Z /\ p =/= X ) ) <-> ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) )
3130rexbii 3041 . . . . . . . . 9 |- ( E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z /\ p =/= X ) ) <-> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) )
3226, 31sylib 208 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) )
3317, 32lhpexle1lem 35293 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) )
34 an31 841 . . . . . . . . . 10 |- ( ( ( p =/= Y /\ p =/= Z ) /\ p =/= X ) <-> ( ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
3534anbi2i 730 . . . . . . . . 9 |- ( ( p .<_ W /\ ( ( p =/= Y /\ p =/= Z ) /\ p =/= X ) ) <-> ( p .<_ W /\ ( ( p =/= X /\ p =/= Z ) /\ p =/= Y ) ) )
36 3anass 1042 . . . . . . . . 9 |- ( ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) <-> ( p .<_ W /\ ( ( p =/= X /\ p =/= Z ) /\ p =/= Y ) ) )
3735, 29, 363bitr4i 292 . . . . . . . 8 |- ( ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) <-> ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
3837rexbii 3041 . . . . . . 7 |- ( E. p e. A ( p .<_ W /\ ( p =/= Y /\ p =/= Z ) /\ p =/= X ) <-> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
3933, 38sylib 208 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
40393expa 1265 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
4112, 40lhpexle1lem 35293 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) )
42 an32 839 . . . . . . 7 |- ( ( ( p =/= X /\ p =/= Z ) /\ p =/= Y ) <-> ( ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
4342anbi2i 730 . . . . . 6 |- ( ( p .<_ W /\ ( ( p =/= X /\ p =/= Z ) /\ p =/= Y ) ) <-> ( p .<_ W /\ ( ( p =/= X /\ p =/= Y ) /\ p =/= Z ) ) )
44 3anass 1042 . . . . . 6 |- ( ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) <-> ( p .<_ W /\ ( ( p =/= X /\ p =/= Y ) /\ p =/= Z ) ) )
4543, 36, 443bitr4i 292 . . . . 5 |- ( ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) <-> ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
4645rexbii 3041 . . . 4 |- ( E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Z ) /\ p =/= Y ) <-> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
4741, 46sylib 208 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Z e. A /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
487, 47lhpexle1lem 35293 . 2 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
49 df-3an 1039 . . . . 5 |- ( ( p =/= X /\ p =/= Y /\ p =/= Z ) <-> ( ( p =/= X /\ p =/= Y ) /\ p =/= Z ) )
5049anbi2i 730 . . . 4 |- ( ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) <-> ( p .<_ W /\ ( ( p =/= X /\ p =/= Y ) /\ p =/= Z ) ) )
5144, 50bitr4i 267 . . 3 |- ( ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) <-> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
5251rexbii 3041 . 2 |- ( E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y ) /\ p =/= Z ) <-> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
5348, 52sylib 208 1 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274
This theorem is referenced by:  cdlemftr3  35853
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