Theorem hlhgt2 34675

Description: A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)

Hypotheses

Ref	Expression
hlhgt4.b	|- B = ( Base ` K )
hlhgt4.s	|- .< = ( lt ` K )
hlhgt4.z	|- .0. = ( 0. ` K )
hlhgt4.u	|- .1. = ( 1. ` K )

Assertion

Ref	Expression
hlhgt2	|- ( K e. HL -> E. x e. B ( .0. .< x /\ x .< .1. ) )

Distinct variable groups:   x, B   x, K

Allowed substitution hints:   .< ( x)   .1. ( x)   .0. ( x)

Proof of Theorem hlhgt2

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlhgt4.b	. . 3 |- B = ( Base ` K )
2		hlhgt4.s	. . 3 |- .< = ( lt ` K )
3		hlhgt4.z	. . 3 |- .0. = ( 0. ` K )
4		hlhgt4.u	. . 3 |- .1. = ( 1. ` K )
5	1, 2, 3, 4	hlhgt4 34674	. 2 |- ( K e. HL -> E. y e. B E. x e. B E. z e. B ( ( .0. .< y /\ y .< x ) /\ ( x .< z /\ z .< .1. ) ) )
6		hlpos 34652	. . . . . . . 8 |- ( K e. HL -> K e. Poset )
7	6	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> K e. Poset )
8		hlop 34649	. . . . . . . . 9 |- ( K e. HL -> K e. OP )
9	8	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> K e. OP )
10	1, 3	op0cl 34471	. . . . . . . 8 |- ( K e. OP -> .0. e. B )
11	9, 10	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> .0. e. B )
12		simpllr 799	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> y e. B )
13		simplr 792	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> x e. B )
14	1, 2	plttr 16970	. . . . . . 7 |- ( ( K e. Poset /\ ( .0. e. B /\ y e. B /\ x e. B ) ) -> ( ( .0. .< y /\ y .< x ) -> .0. .< x ) )
15	7, 11, 12, 13, 14	syl13anc 1328	. . . . . 6 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> ( ( .0. .< y /\ y .< x ) -> .0. .< x ) )
16		simpr 477	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> z e. B )
17	1, 4	op1cl 34472	. . . . . . . 8 |- ( K e. OP -> .1. e. B )
18	9, 17	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> .1. e. B )
19	1, 2	plttr 16970	. . . . . . 7 |- ( ( K e. Poset /\ ( x e. B /\ z e. B /\ .1. e. B ) ) -> ( ( x .< z /\ z .< .1. ) -> x .< .1. ) )
20	7, 13, 16, 18, 19	syl13anc 1328	. . . . . 6 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> ( ( x .< z /\ z .< .1. ) -> x .< .1. ) )
21	15, 20	anim12d 586	. . . . 5 |- ( ( ( ( K e. HL /\ y e. B ) /\ x e. B ) /\ z e. B ) -> ( ( ( .0. .< y /\ y .< x ) /\ ( x .< z /\ z .< .1. ) ) -> ( .0. .< x /\ x .< .1. ) ) )
22	21	rexlimdva 3031	. . . 4 |- ( ( ( K e. HL /\ y e. B ) /\ x e. B ) -> ( E. z e. B ( ( .0. .< y /\ y .< x ) /\ ( x .< z /\ z .< .1. ) ) -> ( .0. .< x /\ x .< .1. ) ) )
23	22	reximdva 3017	. . 3 |- ( ( K e. HL /\ y e. B ) -> ( E. x e. B E. z e. B ( ( .0. .< y /\ y .< x ) /\ ( x .< z /\ z .< .1. ) ) -> E. x e. B ( .0. .< x /\ x .< .1. ) ) )
24	23	rexlimdva 3031	. 2 |- ( K e. HL -> ( E. y e. B E. x e. B E. z e. B ( ( .0. .< y /\ y .< x ) /\ ( x .< z /\ z .< .1. ) ) -> E. x e. B ( .0. .< x /\ x .< .1. ) ) )
25	5, 24	mpd 15	1 |- ( K e. HL -> E. x e. B ( .0. .< x /\ x .< .1. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   Posetcpo 16940   ltcplt 16941   0.cp0 17037   1.cp1 17038   OPcops 34459   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

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-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-p0 17039  df-p1 17040  df-lat 17046  df-oposet 34463  df-ol 34465  df-oml 34466  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  hl0lt1N  34676  hl2at  34691