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Theorem hl2at 34691
Description: A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
Hypothesis
Ref Expression
hl2atom.a |- A = ( Atoms ` K )
Assertion
Ref Expression
hl2at |- ( K e. HL -> E. p e. A E. q e. A p =/= q )
Distinct variable groups:   q, p, A   K, p, q
Proof of Theorem hl2at
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqid 2622 . . 3 |- ( Base ` K ) = ( Base ` K )
2 eqid 2622 . . 3 |- ( lt ` K ) = ( lt ` K )
3 eqid 2622 . . 3 |- ( 0. ` K ) = ( 0. ` K )
4 eqid 2622 . . 3 |- ( 1. ` K ) = ( 1. ` K )
51, 2, 3, 4hlhgt2 34675 . 2 |- ( K e. HL -> E. x e. ( Base ` K ) ( ( 0. ` K ) ( lt ` K ) x /\ x ( lt ` K ) ( 1. ` K ) ) )
6 simpl 473 . . . . . 6 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> K e. HL )
7 hlop 34649 . . . . . . . 8 |- ( K e. HL -> K e. OP )
87adantr 481 . . . . . . 7 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> K e. OP )
91, 3op0cl 34471 . . . . . . 7 |- ( K e. OP -> ( 0. ` K ) e. ( Base ` K ) )
108, 9syl 17 . . . . . 6 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( 0. ` K ) e. ( Base ` K ) )
11 simpr 477 . . . . . 6 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> x e. ( Base ` K ) )
12 eqid 2622 . . . . . . 7 |- ( le ` K ) = ( le ` K )
13 hl2atom.a . . . . . . 7 |- A = ( Atoms ` K )
141, 12, 2, 13hlrelat1 34686 . . . . . 6 |- ( ( K e. HL /\ ( 0. ` K ) e. ( Base ` K ) /\ x e. ( Base ` K ) ) -> ( ( 0. ` K ) ( lt ` K ) x -> E. p e. A ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) ) )
156, 10, 11, 14syl3anc 1326 . . . . 5 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( ( 0. ` K ) ( lt ` K ) x -> E. p e. A ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) ) )
161, 4op1cl 34472 . . . . . . 7 |- ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
178, 16syl 17 . . . . . 6 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( 1. ` K ) e. ( Base ` K ) )
181, 12, 2, 13hlrelat1 34686 . . . . . 6 |- ( ( K e. HL /\ x e. ( Base ` K ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( x ( lt ` K ) ( 1. ` K ) -> E. q e. A ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) )
1917, 18mpd3an3 1425 . . . . 5 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( x ( lt ` K ) ( 1. ` K ) -> E. q e. A ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) )
2015, 19anim12d 586 . . . 4 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( ( ( 0. ` K ) ( lt ` K ) x /\ x ( lt ` K ) ( 1. ` K ) ) -> ( E. p e. A ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ E. q e. A ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) ) )
21 reeanv 3107 . . . . 5 |- ( E. p e. A E. q e. A ( ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) <-> ( E. p e. A ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ E. q e. A ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) )
22 nbrne2 4673 . . . . . . . 8 |- ( ( p ( le ` K ) x /\ -. q ( le ` K ) x ) -> p =/= q )
2322ad2ant2lr 784 . . . . . . 7 |- ( ( ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) -> p =/= q )
2423reximi 3011 . . . . . 6 |- ( E. q e. A ( ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) -> E. q e. A p =/= q )
2524reximi 3011 . . . . 5 |- ( E. p e. A E. q e. A ( ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) -> E. p e. A E. q e. A p =/= q )
2621, 25sylbir 225 . . . 4 |- ( ( E. p e. A ( -. p ( le ` K ) ( 0. ` K ) /\ p ( le ` K ) x ) /\ E. q e. A ( -. q ( le ` K ) x /\ q ( le ` K ) ( 1. ` K ) ) ) -> E. p e. A E. q e. A p =/= q )
2720, 26syl6 35 . . 3 |- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( ( ( 0. ` K ) ( lt ` K ) x /\ x ( lt ` K ) ( 1. ` K ) ) -> E. p e. A E. q e. A p =/= q ) )
2827rexlimdva 3031 . 2 |- ( K e. HL -> ( E. x e. ( Base ` K ) ( ( 0. ` K ) ( lt ` K ) x /\ x ( lt ` K ) ( 1. ` K ) ) -> E. p e. A E. q e. A p =/= q ) )
295, 28mpd 15 1 |- ( K e. HL -> E. p e. A E. q e. A p =/= q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   0.cp0 17037   1.cp1 17038   OPcops 34459   Atomscatm 34550   HLchlt 34637
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
This theorem is referenced by:  atex  34692
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