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Theorem tgqioo 22603

Description: The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)

**Hypothesis**
Ref	Expression
tgqioo.1

**Assertion**
Ref	Expression
tgqioo

Proof of Theorem tgqioo Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgqioo.1	. 2
2		imassrn 5477	. . 3
3		ioof 12271	. . . . . 6
4		ffn 6045	. . . . . 6
5	3, 4	ax-mp 5	. . . . 5
6		simpll 790	. . . . . . . . . 10 $/\$ $/\$
7		elioo1 12215	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
8	7	biimpa 501	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
9	8	simp1d 1073	. . . . . . . . . 10 $/\$ $/\$
10	8	simp2d 1074	. . . . . . . . . 10 $/\$ $/\$
11		qbtwnxr 12031	. . . . . . . . . 10 $/\$ $/\$ $/\$
12	6, 9, 10, 11	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
13		simplr 792	. . . . . . . . . 10 $/\$ $/\$
14	8	simp3d 1075	. . . . . . . . . 10 $/\$ $/\$
15		qbtwnxr 12031	. . . . . . . . . 10 $/\$ $/\$ $/\$
16	9, 13, 14, 15	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$
17		reeanv 3107	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		df-ov 6653	. . . . . . . . . . . . . 14
19		opelxpi 5148	. . . . . . . . . . . . . . . 16 $/\$
20	19	3ad2ant2 1083	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		ffun 6048	. . . . . . . . . . . . . . . . 17
22	3, 21	ax-mp 5	. . . . . . . . . . . . . . . 16
23		qssre 11798	. . . . . . . . . . . . . . . . . . 19
24		ressxr 10083	. . . . . . . . . . . . . . . . . . 19
25	23, 24	sstri 3612	. . . . . . . . . . . . . . . . . 18
26		xpss12 5225	. . . . . . . . . . . . . . . . . 18 $/\$
27	25, 25, 26	mp2an 708	. . . . . . . . . . . . . . . . 17
28	3	fdmi 6052	. . . . . . . . . . . . . . . . 17
29	27, 28	sseqtr4i 3638	. . . . . . . . . . . . . . . 16
30		funfvima2 6493	. . . . . . . . . . . . . . . 16 $/\$
31	22, 29, 30	mp2an 708	. . . . . . . . . . . . . . 15
32	20, 31	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	18, 32	syl5eqel 2705	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	9	3ad2ant1 1082	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simp3lr 1133	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simp3rl 1134	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp2l 1087	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	25, 37	sseldi 3601	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2r 1088	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	25, 39	sseldi 3601	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		elioo1 12215	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
42	38, 40, 41	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	34, 35, 36, 42	mpbir3and 1245	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	6	3ad2ant1 1082	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp3ll 1132	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46		xrltle 11982	. . . . . . . . . . . . . . . . 17 $/\$
47	44, 38, 46	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	45, 47	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 12210	. . . . . . . . . . . . . . 15 $/\$
50	44, 48, 49	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	13	3ad2ant1 1082	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1135	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		xrltle 11982	. . . . . . . . . . . . . . . . 17 $/\$
54	40, 51, 53	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 54	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		iooss2 12211	. . . . . . . . . . . . . . 15 $/\$
57	51, 55, 56	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	50, 57	sstrd 3613	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59		eleq2 2690	. . . . . . . . . . . . . . 15
60		sseq1 3626	. . . . . . . . . . . . . . 15
61	59, 60	anbi12d 747	. . . . . . . . . . . . . 14 $/\$ $/\$
62	61	rspcev 3309	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	33, 43, 58, 62	syl12anc 1324	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3exp 1264	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	64	rexlimdvv 3037	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	17, 65	syl5bir 233	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	12, 16, 66	mp2and 715	. . . . . . . 8 $/\$ $/\$ $/\$
68	67	ralrimiva 2966	. . . . . . 7 $/\$ $/\$
69		qtopbas 22563	. . . . . . . 8
70		eltg2b 20763	. . . . . . . 8 $/\$
71	69, 70	ax-mp 5	. . . . . . 7 $/\$
72	68, 71	sylibr 224	. . . . . 6 $/\$
73	72	rgen2a 2977	. . . . 5
74		ffnov 6764	. . . . 5 $/\$
75	5, 73, 74	mpbir2an 955	. . . 4
76		frn 6053	. . . 4
77	75, 76	ax-mp 5	. . 3
78		2basgen 20794	. . 3 $/\$
79	2, 77, 78	mp2an 708	. 2
80	1, 79	eqtr2i 2645	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wral 2912

wrex 2913

wss 3574

cpw 4158

cop 4183 class class class wbr 4653

cxp 5112

cdm 5114

crn 5115

cima 5117

wfun 5882

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cr 9935

cxr 10073

clt 10074

cle 10075

cq 11788

cioo 12175

ctg 16098

ctb 20749

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-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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-topgen 16104 df-bases 20750

This theorem is referenced by: re2ndc 22604 opnmblALT 23371 mbfimaopnlem 23422 tgqioo2 39774

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