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Theorem tgqioo 14524

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 9939	. . . 4
3	2	imaex 5001	. . 3
4		imassrn 4999	. . 3
5		ioof 10003	. . . . . 6
6		ffn 5384	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 527	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 9943	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 296	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 1011	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1012	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10290	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 528	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1013	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10290	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2660	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 5900	. . . . . . . . . . . . . 14
21		opelxpi 4676	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1021	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5387	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 9662	. . . . . . . . . . . . . . . . . . 19
26		ressxr 8032	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3179	. . . . . . . . . . . . . . . . . 18
28		xpss12 4751	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 426	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5392	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3205	. . . . . . . . . . . . . . . 16
32		funfvima2 5770	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 426	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1020	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1071	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1072	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1025	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3168	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1026	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3168	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 9943	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
44	40, 42, 43	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	36, 37, 38, 44	mpbir3and 1182	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	8	3ad2ant1 1020	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp3ll 1070	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	46, 40, 47	xrltled 9831	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 9948	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1020	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1073	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9831	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 9949	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3180	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2253	. . . . . . . . . . . . . . 15
58		sseq1 3193	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 473	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2856	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1247	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1204	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2614	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	biimtrrid 153	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 433	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2563	. . . . . . 7 $/\$ $/\$
67		qtopbas 14499	. . . . . . . 8
68		eltg2b 14031	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 134	. . . . . 6 $/\$
71	70	rgen2a 2544	. . . . 5
72		ffnov 6001	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 944	. . . 4
74		frn 5393	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 14059	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1348	. 2
78	1, 77	eqtr2i 2211	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2160

wral 2468

wrex 2469

cvv 2752

wss 3144

cpw 3590

cop 3610 class class class wbr 4018

cxp 4642

cdm 4644

crn 4645

cima 4647

wfun 5229

wfn 5230

wf 5231

cfv 5235 (class class class)co 5897

cr 7841

cxr 8022

clt 8023

cle 8024

cq 9651

cioo 9920

ctg 12762

ctb 14019

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-sup 7014 df-inf 7015 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-xneg 9804 df-ioo 9924 df-seqfrec 10479 df-exp 10554 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-topgen 12768 df-bases 14020

This theorem is referenced by: (None)

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