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Theorem tgqioo 14791

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 9982	. . . 4
3	2	imaex 5024	. . 3
4		imassrn 5020	. . 3
5		ioof 10046	. . . . . 6
6		ffn 5407	. . . . . 6
7	5, 6	ax-mp 5	. . . . 5
8		simpll 527	. . . . . . . . . 10 $/\$ $/\$
9		elioo1 9986	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
10	9	biimpa 296	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11	10	simp1d 1011	. . . . . . . . . 10 $/\$ $/\$
12	10	simp2d 1012	. . . . . . . . . 10 $/\$ $/\$
13		qbtwnxr 10347	. . . . . . . . . 10 $/\$ $/\$ $/\$
14	8, 11, 12, 13	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
15		simplr 528	. . . . . . . . . 10 $/\$ $/\$
16	10	simp3d 1013	. . . . . . . . . 10 $/\$ $/\$
17		qbtwnxr 10347	. . . . . . . . . 10 $/\$ $/\$ $/\$
18	11, 15, 16, 17	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$
19		reeanv 2667	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		df-ov 5925	. . . . . . . . . . . . . 14
21		opelxpi 4695	. . . . . . . . . . . . . . . 16 $/\$
22	21	3ad2ant2 1021	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ffun 5410	. . . . . . . . . . . . . . . . 17
24	5, 23	ax-mp 5	. . . . . . . . . . . . . . . 16
25		qssre 9704	. . . . . . . . . . . . . . . . . . 19
26		ressxr 8070	. . . . . . . . . . . . . . . . . . 19
27	25, 26	sstri 3192	. . . . . . . . . . . . . . . . . 18
28		xpss12 4770	. . . . . . . . . . . . . . . . . 18 $/\$
29	27, 27, 28	mp2an 426	. . . . . . . . . . . . . . . . 17
30	5	fdmi 5415	. . . . . . . . . . . . . . . . 17
31	29, 30	sseqtrri 3218	. . . . . . . . . . . . . . . 16
32		funfvima2 5795	. . . . . . . . . . . . . . . 16 $/\$
33	24, 31, 32	mp2an 426	. . . . . . . . . . . . . . 15
34	22, 33	syl 14	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 34	eqeltrid 2283	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	11	3ad2ant1 1020	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3lr 1071	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3rl 1072	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2l 1025	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	27, 39	sselid 3181	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simp2r 1026	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	27, 41	sselid 3181	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		elioo1 9986	. . . . . . . . . . . . . . 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 9874	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 9991	. . . . . . . . . . . . . . 15 $/\$
50	46, 48, 49	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	15	3ad2ant1 1020	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1073	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	42, 51, 52	xrltled 9874	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		iooss2 9992	. . . . . . . . . . . . . . 15 $/\$
55	51, 53, 54	syl2anc 411	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	50, 55	sstrd 3193	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		eleq2 2260	. . . . . . . . . . . . . . 15
58		sseq1 3206	. . . . . . . . . . . . . . 15
59	57, 58	anbi12d 473	. . . . . . . . . . . . . 14 $/\$ $/\$
60	59	rspcev 2868	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61	35, 45, 56, 60	syl12anc 1247	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	3exp 1204	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	rexlimdvv 2621	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	19, 63	biimtrrid 153	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	14, 18, 64	mp2and 433	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ralrimiva 2570	. . . . . . 7 $/\$ $/\$
67		qtopbas 14758	. . . . . . . 8
68		eltg2b 14290	. . . . . . . 8 $/\$
69	67, 68	ax-mp 5	. . . . . . 7 $/\$
70	66, 69	sylibr 134	. . . . . 6 $/\$
71	70	rgen2a 2551	. . . . 5
72		ffnov 6026	. . . . 5 $/\$
73	7, 71, 72	mpbir2an 944	. . . 4
74		frn 5416	. . . 4
75	73, 74	ax-mp 5	. . 3
76		2basgeng 14318	. . 3 $/\$ $/\$
77	3, 4, 75, 76	mp3an 1348	. 2
78	1, 77	eqtr2i 2218	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2167

wral 2475

wrex 2476

cvv 2763

wss 3157

cpw 3605

cop 3625 class class class wbr 4033

cxp 4661

cdm 4663

crn 4664

cima 4666

wfun 5252

wfn 5253

wf 5254

cfv 5258 (class class class)co 5922

cr 7878

cxr 8060

clt 8061

cle 8062

cq 9693

cioo 9963

ctg 12925

ctb 14278

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-xneg 9847 df-ioo 9967 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-topgen 12931 df-bases 14279

This theorem is referenced by: (None)

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