Theorem lhpexle1 35294

Description: There exists an atom under a co-atom different from any given element. (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
lhpexle1	|- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X ) )

Distinct variable groups:   .<_ , p   A, p   H, p   K, p   W, p   X, p

Proof of Theorem lhpexle1

Step	Hyp	Ref	Expression
1		lhpex1.l	. . . . 5 |- .<_ = ( le ` K )
2		lhpex1.a	. . . . 5 |- A = ( Atoms ` K )
3		lhpex1.h	. . . . 5 |- H = ( LHyp ` K )
4	1, 2, 3	lhpexle 35291	. . . 4 |- ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W )
5		tru 1487	. . . . . 6 |- T.
6	5	jctr 565	. . . . 5 |- ( p .<_ W -> ( p .<_ W /\ T. ) )
7	6	reximi 3011	. . . 4 |- ( E. p e. A p .<_ W -> E. p e. A ( p .<_ W /\ T. ) )
8	4, 7	syl 17	. . 3 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ T. ) )
9		simpll 790	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> K e. HL )
10		simprl 794	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> X e. A )
11		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
12	11, 3	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. ( Base ` K ) )
13	12	ad2antlr 763	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> W e. ( Base ` K ) )
14		eqid 2622	. . . . . 6 |- ( lt ` K ) = ( lt ` K )
15	1, 14, 2, 3	lhplt 35286	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> X ( lt ` K ) W )
16	11, 14, 2	2atlt 34725	. . . . 5 |- ( ( ( K e. HL /\ X e. A /\ W e. ( Base ` K ) ) /\ X ( lt ` K ) W ) -> E. p e. A ( p =/= X /\ p ( lt ` K ) W ) )
17	9, 10, 13, 15, 16	syl31anc 1329	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p =/= X /\ p ( lt ` K ) W ) )
18		simp3r 1090	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p ( lt ` K ) W )
19		simp1ll 1124	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> K e. HL )
20		simp2 1062	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p e. A )
21		simp1lr 1125	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> W e. H )
22	1, 14	pltle 16961	. . . . . . . . 9 |- ( ( K e. HL /\ p e. A /\ W e. H ) -> ( p ( lt ` K ) W -> p .<_ W ) )
23	19, 20, 21, 22	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> ( p ( lt ` K ) W -> p .<_ W ) )
24	18, 23	mpd 15	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p .<_ W )
25		a1tru 1500	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> T. )
26		simp3l 1089	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p =/= X )
27	24, 25, 26	3jca 1242	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> ( p .<_ W /\ T. /\ p =/= X ) )
28	27	3expia 1267	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A ) -> ( ( p =/= X /\ p ( lt ` K ) W ) -> ( p .<_ W /\ T. /\ p =/= X ) ) )
29	28	reximdva 3017	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> ( E. p e. A ( p =/= X /\ p ( lt ` K ) W ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) ) )
30	17, 29	mpd 15	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) )
31	8, 30	lhpexle1lem 35293	. 2 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) )
32		3simpb 1059	. . 3 |- ( ( p .<_ W /\ T. /\ p =/= X ) -> ( p .<_ W /\ p =/= X ) )
33	32	reximi 3011	. 2 |- ( E. p e. A ( p .<_ W /\ T. /\ p =/= X ) -> E. p e. A ( p .<_ W /\ p =/= X ) )
34	31, 33	syl 17	1 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   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:  lhpexle2lem  35295  lhpexle2  35296  lhpex2leN  35299