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Theorem tgqioo 15060

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 10031	. . . 4
3	2	imaex 5038	. . 3
4		imassrn 5034	. . 3
5		ioof 10095	. . . . . 6
6		ffn 5427	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 527	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 10035	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 296	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 1012	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1013	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10402	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1250	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 528	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1014	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10402	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1250	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2676	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 5949	. . . . . . . . . . . . . 14
21		opelxpi 4708	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1022	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5430	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 9753	. . . . . . . . . . . . . . . . . . 19
26		ressxr 8118	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3202	. . . . . . . . . . . . . . . . . 18
28		xpss12 4783	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 426	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5435	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3228	. . . . . . . . . . . . . . . 16
32		funfvima2 5819	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 426	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2292	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1021	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1072	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1073	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1026	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3191	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1027	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3191	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 10035	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1183	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 1021	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1071	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 9923	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 10040	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1021	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1074	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9923	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 10041	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3203	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2269	. . . . . . . . . . . . . . 15
58		sseq1 3216	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 473	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2877	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1248	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1205	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2630	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	biimtrrid 153	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 433	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2579	. . . . . . 7 $/\$ $/\$
67		qtopbas 15027	. . . . . . . 8
68		eltg2b 14559	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 134	. . . . . 6 $/\$
71	70	rgen2a 2560	. . . . 5
72		ffnov 6051	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 945	. . . 4
74		frn 5436	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 14587	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1350	. 2
78	1, 77	eqtr2i 2227	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 981

wceq 1373

wcel 2176

wral 2484

wrex 2485

cvv 2772

wss 3166

cpw 3616

cop 3636 class class class wbr 4045

cxp 4674

cdm 4676

crn 4677

cima 4679

wfun 5266

wfn 5267

wf 5268

cfv 5272 (class class class)co 5946

cr 7926

cxr 8108

clt 8109

cle 8110

cq 9742

cioo 10012

ctg 13119

ctb 14547

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-isom 5281 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-frec 6479 df-sup 7088 df-inf 7089 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-xneg 9896 df-ioo 10016 df-seqfrec 10595 df-exp 10686 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-topgen 13125 df-bases 14548

This theorem is referenced by: (None)

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