reconnlem1 - Metamath Proof Explorer

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Theorem reconnlem1 22629

Description: Lemma for reconn 22631. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)

**Assertion**
Ref	Expression
reconnlem1	$/\$ ↾_t Conn $/\$ $/\$

Proof of Theorem reconnlem1 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . 4 $/\$ ↾_t Conn $/\$ $/\$ ↾_t Conn
2		retopon 22567	. . . . . . 7 TopOn
3	2	a1i 11	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ TopOn
4		simplll 798	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
5		iooretop 22569	. . . . . . 7
6	5	a1i 11	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
7		iooretop 22569	. . . . . . 7
8	7	a1i 11	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
9		simplrl 800	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
10	4, 9	sseldd 3604	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
11		mnflt 11957	. . . . . . . . 9
12	10, 11	syl 17	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
13		eldifn 3733	. . . . . . . . . . 11 $\$
14	13	adantl 482	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
15		eleq1 2689	. . . . . . . . . . 11
16	9, 15	syl5ibcom 235	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
17	14, 16	mtod 189	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
18		eldifi 3732	. . . . . . . . . . . . . 14 $\$
19	18	adantl 482	. . . . . . . . . . . . 13 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
20		simplrr 801	. . . . . . . . . . . . . . 15 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
21	4, 20	sseldd 3604	. . . . . . . . . . . . . 14 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
22		elicc2 12238	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
23	10, 21, 22	syl2anc 693	. . . . . . . . . . . . 13 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$ $/\$
24	19, 23	mpbid 222	. . . . . . . . . . . 12 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$ $/\$
25	24	simp2d 1074	. . . . . . . . . . 11 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
26	24	simp1d 1073	. . . . . . . . . . . 12 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
27	10, 26	leloed 10180	. . . . . . . . . . 11 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $\/$
28	25, 27	mpbid 222	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $\/$
29	28	ord 392	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
30	17, 29	mt3d 140	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
31		mnfxr 10096	. . . . . . . . 9
32	26	rexrd 10089	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
33		elioo2 12216	. . . . . . . . 9 $/\$ $/\$ $/\$
34	31, 32, 33	sylancr 695	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$ $/\$
35	10, 12, 30, 34	mpbir3and 1245	. . . . . . 7 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
36		inelcm 4032	. . . . . . 7 $/\$
37	35, 9, 36	syl2anc 693	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
38		eleq1 2689	. . . . . . . . . . 11
39	20, 38	syl5ibrcom 237	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
40	14, 39	mtod 189	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
41	24	simp3d 1075	. . . . . . . . . . 11 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
42	26, 21	leloed 10180	. . . . . . . . . . 11 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $\/$
43	41, 42	mpbid 222	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $\/$
44	43	ord 392	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
45	40, 44	mt3d 140	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
46		ltpnf 11954	. . . . . . . . 9
47	21, 46	syl 17	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
48		pnfxr 10092	. . . . . . . . 9
49		elioo2 12216	. . . . . . . . 9 $/\$ $/\$ $/\$
50	32, 48, 49	sylancl 694	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$ $/\$
51	21, 45, 47, 50	mpbir3and 1245	. . . . . . 7 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
52		inelcm 4032	. . . . . . 7 $/\$
53	51, 20, 52	syl2anc 693	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
54		inss1 3833	. . . . . . 7
55	32, 31	jctil 560	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$
56	32, 48	jctir 561	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ $/\$
57	26	leidd 10594	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
58		ioodisj 12302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
59	55, 56, 57, 58	syl21anc 1325	. . . . . . 7 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
60		sseq0 3975	. . . . . . 7 $/\$
61	54, 59, 60	sylancr 695	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
62	31	a1i 11	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
63	48	a1i 11	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
64		mnflt 11957	. . . . . . . . . . 11
65	26, 64	syl 17	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
66		ltpnf 11954	. . . . . . . . . . 11
67	26, 66	syl 17	. . . . . . . . . 10 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
68		ioojoin 12303	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69	62, 32, 63, 65, 67, 68	syl32anc 1334	. . . . . . . . 9 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
70		unass 3770	. . . . . . . . . 10
71		un12 3771	. . . . . . . . . 10
72	70, 71	eqtri 2644	. . . . . . . . 9
73		ioomax 12248	. . . . . . . . 9
74	69, 72, 73	3eqtr3g 2679	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
75	4, 74	sseqtr4d 3642	. . . . . . 7 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
76		disjsn 4246	. . . . . . . . 9
77	14, 76	sylibr 224	. . . . . . . 8 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
78		disjssun 4036	. . . . . . . 8
79	77, 78	syl 17	. . . . . . 7 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
80	75, 79	mpbid 222	. . . . . 6 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$
81	3, 4, 6, 8, 37, 53, 61, 80	nconnsubb 21226	. . . . 5 $/\$ ↾_t Conn $/\$ $/\$ $/\$ $\$ ↾_t Conn
82	81	ex 450	. . . 4 $/\$ ↾_t Conn $/\$ $/\$ $\$ ↾_t Conn
83	1, 82	mt2d 131	. . 3 $/\$ ↾_t Conn $/\$ $/\$ $\$
84	83	eq0rdv 3979	. 2 $/\$ ↾_t Conn $/\$ $/\$ $\$
85		ssdif0 3942	. 2 $\$
86	84, 85	sylibr 224	1 $/\$ ↾_t Conn $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

csn 4177 class class class wbr 4653

crn 5115

cfv 5888 (class class class)co 6650

cr 9935

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cle 10075

cioo 12175

cicc 12178 ↾_t crest 16081

ctg 16098 TopOnctopon 20715 Conncconn 21214

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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ioo 12179 df-ico 12181 df-icc 12182 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-conn 21215

This theorem is referenced by: reconn 22631

W3C validator