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Theorem tgqioo 12466

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 9531	. . . 4
3	2	imaex 4830	. . 3
4		imassrn 4828	. . 3
5		ioof 9595	. . . . . 6
6		ffn 5208	. . . . . 6
7	5, 6	ax-mp 7	. . . . 5
8		simpll 499	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 9535	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 292	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 961	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 962	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 9876	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1184	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 500	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 963	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 9876	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1184	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2558	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 5709	. . . . . . . . . . . . . 14
21		opelxpi 4509	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 971	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5211	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 7	. . . . . . . . . . . . . . . 16
25		qssre 9272	. . . . . . . . . . . . . . . . . . 19
26		ressxr 7681	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3056	. . . . . . . . . . . . . . . . . 18
28		xpss12 4584	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 420	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5216	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtr4i 3082	. . . . . . . . . . . . . . . 16
32		funfvima2 5582	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 420	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	syl5eqel 2186	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 970	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1021	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1022	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 975	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sseldi 3045	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 976	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sseldi 3045	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 9535	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 406	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1132	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 970	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1020	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 9426	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 9540	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 406	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 970	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1023	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9426	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 9541	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 406	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3057	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2163	. . . . . . . . . . . . . . 15
58		sseq1 3070	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 460	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2744	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1182	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1148	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2515	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	syl5bir 152	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 427	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2464	. . . . . . 7 $/\$ $/\$
67		qtopbas 12444	. . . . . . . 8
68		eltg2b 12005	. . . . . . . 8 $/\$
69	67, 68	ax-mp 7	. . . . . . 7 $/\$
70	66, 69	sylibr 133	. . . . . 6 $/\$
71	70	rgen2a 2445	. . . . 5
72		ffnov 5807	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 894	. . . 4
74		frn 5217	. . . 4
75	73, 74	ax-mp 7	. . 3
76		2basgeng 12033	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1283	. 2
78	1, 77	eqtr2i 2121	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 930

wceq 1299

wcel 1448

wral 2375

wrex 2376

cvv 2641

wss 3021

cpw 3457

cop 3477 class class class wbr 3875

cxp 4475

cdm 4477

crn 4478

cima 4480

wfun 5053

wfn 5054

wf 5055

cfv 5059 (class class class)co 5706

cr 7499

cxr 7671

clt 7672

cle 7673

cq 9261

cioo 9512

ctg 11917

ctb 11991

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-sup 6786 df-inf 6787 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-xneg 9400 df-ioo 9516 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-topgen 11923 df-bases 11992

This theorem is referenced by: (None)

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