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Theorem tgqioo 18784

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 5175	. . 3
3		ioof 10958	. . . . . 6
4		ffn 5550	. . . . . 6
5	3, 4	ax-mp 8	. . . . 5
6		simpll 731	. . . . . . . . . 10 $/\$ $/\$
7		elioo1 10912	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
8	7	biimpa 471	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
9	8	simp1d 969	. . . . . . . . . 10 $/\$ $/\$
10	8	simp2d 970	. . . . . . . . . 10 $/\$ $/\$
11		qbtwnxr 10742	. . . . . . . . . 10 $/\$ $/\$ $/\$
12	6, 9, 10, 11	syl3anc 1184	. . . . . . . . 9 $/\$ $/\$ $/\$
13		simplr 732	. . . . . . . . . 10 $/\$ $/\$
14	8	simp3d 971	. . . . . . . . . 10 $/\$ $/\$
15		qbtwnxr 10742	. . . . . . . . . 10 $/\$ $/\$ $/\$
16	9, 13, 14, 15	syl3anc 1184	. . . . . . . . 9 $/\$ $/\$ $/\$
17		reeanv 2835	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		df-ov 6043	. . . . . . . . . . . . . 14
19		opelxpi 4869	. . . . . . . . . . . . . . . 16 $/\$
20	19	3ad2ant2 979	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		ffun 5552	. . . . . . . . . . . . . . . . 17
22	3, 21	ax-mp 8	. . . . . . . . . . . . . . . 16
23		qssre 10540	. . . . . . . . . . . . . . . . . . 19
24		ressxr 9085	. . . . . . . . . . . . . . . . . . 19
25	23, 24	sstri 3317	. . . . . . . . . . . . . . . . . 18
26		xpss12 4940	. . . . . . . . . . . . . . . . . 18 $/\$
27	25, 25, 26	mp2an 654	. . . . . . . . . . . . . . . . 17
28	3	fdmi 5555	. . . . . . . . . . . . . . . . 17
29	27, 28	sseqtr4i 3341	. . . . . . . . . . . . . . . 16
30		funfvima2 5933	. . . . . . . . . . . . . . . 16 $/\$
31	22, 29, 30	mp2an 654	. . . . . . . . . . . . . . 15
32	20, 31	syl 16	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	18, 32	syl5eqel 2488	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	9	3ad2ant1 978	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simp3lr 1029	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simp3rl 1030	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp2l 983	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	25, 37	sseldi 3306	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2r 984	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	25, 39	sseldi 3306	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		elioo1 10912	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
42	38, 40, 41	syl2anc 643	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	34, 35, 36, 42	mpbir3and 1137	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	6	3ad2ant1 978	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp3ll 1028	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46		xrltle 10698	. . . . . . . . . . . . . . . . 17 $/\$
47	44, 38, 46	syl2anc 643	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	45, 47	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 10907	. . . . . . . . . . . . . . 15 $/\$
50	44, 48, 49	syl2anc 643	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	13	3ad2ant1 978	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1031	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		xrltle 10698	. . . . . . . . . . . . . . . . 17 $/\$
54	40, 51, 53	syl2anc 643	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 54	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		iooss2 10908	. . . . . . . . . . . . . . 15 $/\$
57	51, 55, 56	syl2anc 643	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	50, 57	sstrd 3318	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59		eleq2 2465	. . . . . . . . . . . . . . 15
60		sseq1 3329	. . . . . . . . . . . . . . 15
61	59, 60	anbi12d 692	. . . . . . . . . . . . . 14 $/\$ $/\$
62	61	rspcev 3012	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	33, 43, 58, 62	syl12anc 1182	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3exp 1152	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	64	rexlimdvv 2796	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	17, 65	syl5bir 210	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	12, 16, 66	mp2and 661	. . . . . . . 8 $/\$ $/\$ $/\$
68	67	ralrimiva 2749	. . . . . . 7 $/\$ $/\$
69		qtopbas 18746	. . . . . . . 8
70		eltg2b 16979	. . . . . . . 8 $/\$
71	69, 70	ax-mp 8	. . . . . . 7 $/\$
72	68, 71	sylibr 204	. . . . . 6 $/\$
73	72	rgen2a 2732	. . . . 5
74		ffnov 6133	. . . . 5 $/\$
75	5, 73, 74	mpbir2an 887	. . . 4
76		frn 5556	. . . 4
77	75, 76	ax-mp 8	. . 3
78		2basgen 17010	. . 3 $/\$
79	2, 77, 78	mp2an 654	. 2
80	1, 79	eqtr2i 2425	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 177 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wral 2666

wrex 2667

wss 3280

cpw 3759

cop 3777 class class class wbr 4172

cxp 4835

cdm 4837

crn 4838

cima 4840

wfun 5407

wfn 5408

wf 5409

cfv 5413 (class class class)co 6040

cr 8945

cxr 9075

clt 9076

cle 9077

cq 10530

cioo 10872

ctg 13620

ctb 16917

This theorem is referenced by: re2ndc 18785 opnmblALT 19448 mbfimaopnlem 19500

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 ax-pre-sup 9024

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-pss 3296 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-tp 3782 df-op 3783 df-uni 3976 df-iun 4055 df-br 4173 df-opab 4227 df-mpt 4228 df-tr 4263 df-eprel 4454 df-id 4458 df-po 4463 df-so 4464 df-fr 4501 df-we 4503 df-ord 4544 df-on 4545 df-lim 4546 df-suc 4547 df-om 4805 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-riota 6508 df-recs 6592 df-rdg 6627 df-er 6864 df-en 7069 df-dom 7070 df-sdom 7071 df-sup 7404 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-div 9634 df-nn 9957 df-n0 10178 df-z 10239 df-uz 10445 df-q 10531 df-ioo 10876 df-topgen 13622 df-bases 16920

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