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Theorem tgqioo 15781

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 10326	. . . 4
3	2	imaex 5144	. . 3
4		imassrn 5140	. . 3
5		ioof 10390	. . . . . 6
6		ffn 5536	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 531	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 10330	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 296	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 1040	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1041	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10709	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 533	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1042	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10709	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2721	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 6091	. . . . . . . . . . . . . 14
21		opelxpi 4808	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1050	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5539	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 10046	. . . . . . . . . . . . . . . . . . 19
26		ressxr 8375	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3257	. . . . . . . . . . . . . . . . . 18
28		xpss12 4884	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 430	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5544	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3283	. . . . . . . . . . . . . . . 16
32		funfvima2 5954	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 430	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2325	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1049	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1100	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1101	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1054	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3246	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1055	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3246	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 10330	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 415	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1211	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 1049	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1099	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 10218	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 10335	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 415	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1049	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1102	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 10218	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 10336	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 415	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3258	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2302	. . . . . . . . . . . . . . 15
58		sseq1 3271	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 477	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2929	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1233	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2675	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	biimtrrid 153	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 437	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2623	. . . . . . 7 $/\$ $/\$
67		qtopbas 15748	. . . . . . . 8
68		eltg2b 15280	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 134	. . . . . 6 $/\$
71	70	rgen2a 2604	. . . . 5
72		ffnov 6195	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 955	. . . 4
74		frn 5545	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 15308	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1378	. 2
78	1, 77	eqtr2i 2260	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209

wral 2528

wrex 2529

cvv 2821

wss 3220

cpw 3688

cop 3712 class class class wbr 4131

cxp 4774

cdm 4776

crn 4777

cima 4779

wfun 5374

wfn 5375

wf 5376

cfv 5380 (class class class)co 6088

cr 8184

cxr 8365

clt 8366

cle 8367

cq 10035

cioo 10307

ctg 13667

ctb 15268

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-ioo 10311 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-topgen 13673 df-bases 15269

This theorem is used by: (None)

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