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Theorem tgqioo 13187

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 9843	. . . 4
3	2	imaex 4959	. . 3
4		imassrn 4957	. . 3
5		ioof 9907	. . . . . 6
6		ffn 5337	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 519	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 9847	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 294	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 999	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1000	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10193	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1228	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 520	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1001	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10193	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1228	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2635	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 5845	. . . . . . . . . . . . . 14
21		opelxpi 4636	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1009	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5340	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 9568	. . . . . . . . . . . . . . . . . . 19
26		ressxr 7942	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3151	. . . . . . . . . . . . . . . . . 18
28		xpss12 4711	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 423	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5345	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3177	. . . . . . . . . . . . . . . 16
32		funfvima2 5717	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 423	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2253	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1008	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1059	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1060	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1013	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3140	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1014	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3140	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 9847	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 409	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1170	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 1008	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1058	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 9735	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 9852	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 409	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1008	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1061	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9735	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 9853	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 409	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3152	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2230	. . . . . . . . . . . . . . 15
58		sseq1 3165	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 465	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2830	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1226	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1192	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2590	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	syl5bir 152	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 430	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2539	. . . . . . 7 $/\$ $/\$
67		qtopbas 13162	. . . . . . . 8
68		eltg2b 12694	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 133	. . . . . 6 $/\$
71	70	rgen2a 2520	. . . . 5
72		ffnov 5946	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 932	. . . 4
74		frn 5346	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 12722	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1327	. 2
78	1, 77	eqtr2i 2187	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 968

wceq 1343

wcel 2136

wral 2444

wrex 2445

cvv 2726

wss 3116

cpw 3559

cop 3579 class class class wbr 3982

cxp 4602

cdm 4604

crn 4605

cima 4607

wfun 5182

wfn 5183

wf 5184

cfv 5188 (class class class)co 5842

cr 7752

cxr 7932

clt 7933

cle 7934

cq 9557

cioo 9824

ctg 12571

ctb 12680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-ioo 9828 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-topgen 12577 df-bases 12681

This theorem is referenced by: (None)

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