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Theorem tgqioo 15229

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		iooex 10103	. . . 4
3	2	imaex 5083	. . 3
4		imassrn 5079	. . 3
5		ioof 10167	. . . . . 6
6		ffn 5473	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 527	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 10107	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 296	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 1033	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1034	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10477	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1271	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 528	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1035	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10477	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1271	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2701	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 6004	. . . . . . . . . . . . . 14
21		opelxpi 4751	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1043	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5476	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 9825	. . . . . . . . . . . . . . . . . . 19
26		ressxr 8190	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3233	. . . . . . . . . . . . . . . . . 18
28		xpss12 4826	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 426	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5481	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3259	. . . . . . . . . . . . . . . 16
32		funfvima2 5872	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 426	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2316	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1042	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1093	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1094	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1047	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1048	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 10107	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1204	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 1042	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1092	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 9995	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 10112	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1042	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1095	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9995	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 10113	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3234	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2293	. . . . . . . . . . . . . . 15
58		sseq1 3247	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 473	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2907	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1269	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1226	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2655	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	biimtrrid 153	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 433	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2603	. . . . . . 7 $/\$ $/\$
67		qtopbas 15196	. . . . . . . 8
68		eltg2b 14728	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 134	. . . . . 6 $/\$
71	70	rgen2a 2584	. . . . 5
72		ffnov 6108	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 948	. . . 4
74		frn 5482	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 14756	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1371	. 2
78	1, 77	eqtr2i 2251	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

wcel 2200

wral 2508

wrex 2509

cvv 2799

wss 3197

cpw 3649

cop 3669 class class class wbr 4083

cxp 4717

cdm 4719

crn 4720

cima 4722

wfun 5312

wfn 5313

wf 5314

cfv 5318 (class class class)co 6001

cr 7998

cxr 8180

clt 8181

cle 8182

cq 9814

cioo 10084

ctg 13287

ctb 14716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-sup 7151 df-inf 7152 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-xneg 9968 df-ioo 10088 df-seqfrec 10670 df-exp 10761 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-topgen 13293 df-bases 14717

This theorem is referenced by: (None)

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