1cvratex - Mathbox for Norm Megill

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Theorem 1cvratex 34759

Description: There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)

**Hypotheses**
Ref	Expression
1cvratex.b
1cvratex.s
1cvratex.u
1cvratex.c
1cvratex.a

**Assertion**
Ref	Expression
1cvratex	$/\$ $/\$

Distinct variable groups:

Proof of Theorem 1cvratex Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 $/\$ $/\$
2		1cvratex.b	. . . . 5
3		1cvratex.u	. . . . 5
4		eqid 2622	. . . . 5
5		1cvratex.c	. . . . 5
6		1cvratex.a	. . . . 5
7	2, 3, 4, 5, 6	1cvrco 34758	. . . 4 $/\$
8	7	biimp3a 1432	. . 3 $/\$ $/\$
9		eqid 2622	. . . 4
10	9, 5, 6	2dim 34756	. . 3 $/\$ $/\$
11	1, 8, 10	syl2anc 693	. 2 $/\$ $/\$ $/\$
12		simp11 1091	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		hlop 34649	. . . . . . . 8
14	12, 13	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		hllat 34650	. . . . . . . . 9
16	12, 15	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		simp12 1092	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	2, 4	opoccl 34481	. . . . . . . . 9 $/\$
19	14, 17, 18	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simp2l 1087	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	2, 6	atbase 34576	. . . . . . . . 9
22	20, 21	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	2, 9	latjcl 17051	. . . . . . . 8 $/\$ $/\$
24	16, 19, 22, 23	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	2, 4	opoccl 34481	. . . . . . 7 $/\$
26	14, 24, 25	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simp2r 1088	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	2, 6	atbase 34576	. . . . . . . . . . . . 13
29	27, 28	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	2, 9	latjcl 17051	. . . . . . . . . . . 12 $/\$ $/\$
31	16, 24, 29, 30	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	2, 4	opoccl 34481	. . . . . . . . . . 11 $/\$
33	14, 31, 32	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		eqid 2622	. . . . . . . . . . 11
35		eqid 2622	. . . . . . . . . . 11
36	2, 34, 35	op0le 34473	. . . . . . . . . 10 $/\$
37	14, 33, 36	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3r 1090	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		1cvratex.s	. . . . . . . . . . . 12
40	2, 39, 5	cvrlt 34557	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	12, 24, 31, 38, 40	syl31anc 1329	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	2, 39, 4	opltcon3b 34491	. . . . . . . . . . 11 $/\$ $/\$
43	14, 24, 31, 42	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 43	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		hlpos 34652	. . . . . . . . . . 11
46	12, 45	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	2, 35	op0cl 34471	. . . . . . . . . . 11
48	14, 47	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	2, 34, 39	plelttr 16972	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
50	46, 48, 33, 26, 49	syl13anc 1328	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	37, 44, 50	mp2and 715	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	39	pltne 16962	. . . . . . . . 9 $/\$ $/\$
53	12, 48, 26, 52	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	mpd 15	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	necomd 2849	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	2, 34, 35, 6	atle 34722	. . . . . 6 $/\$ $/\$
57	12, 26, 55, 56	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		simp3l 1089	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	2, 39, 5	cvrlt 34557	. . . . . . . . . . 11 $/\$ $/\$ $/\$
60	12, 19, 24, 58, 59	syl31anc 1329	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	2, 39, 4	opltcon3b 34491	. . . . . . . . . . 11 $/\$ $/\$
62	14, 19, 24, 61	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	60, 62	mpbid 222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	2, 4	opococ 34482	. . . . . . . . . 10 $/\$
65	14, 17, 64	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	63, 65	breqtrd 4679	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	66	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		simpl11 1136	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68, 45	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	2, 6	atbase 34576	. . . . . . . . 9
71	70	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	26	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		simpl12 1137	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	2, 34, 39	plelttr 16972	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
75	69, 71, 72, 73, 74	syl13anc 1328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	67, 75	mpan2d 710	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	76	reximdva 3017	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	57, 77	mpd 15	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	78	3exp 1264	. . 3 $/\$ $/\$ $/\$ $/\$
80	79	rexlimdvv 3037	. 2 $/\$ $/\$ $/\$
81	11, 80	mpd 15	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

coc 15949

cpo 16940

cplt 16941

cjn 16944

cp0 17037

cp1 17038

clat 17045

cops 34459

ccvr 34549

catm 34550

chlt 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: 1cvratlt 34760 lhpexlt 35288

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