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Theorem tgqioo 21811

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		imassrn 5178	. . 3
3		ioof 11729	. . . . . 6
4		ffn 5726	. . . . . 6
5	3, 4	ax-mp 5	. . . . 5
6		simpll 759	. . . . . . . . . 10 $/\$ $/\$
7		elioo1 11673	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
8	7	biimpa 487	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
9	8	simp1d 1019	. . . . . . . . . 10 $/\$ $/\$
10	8	simp2d 1020	. . . . . . . . . 10 $/\$ $/\$
11		qbtwnxr 11490	. . . . . . . . . 10 $/\$ $/\$ $/\$
12	6, 9, 10, 11	syl3anc 1267	. . . . . . . . 9 $/\$ $/\$ $/\$
13		simplr 761	. . . . . . . . . 10 $/\$ $/\$
14	8	simp3d 1021	. . . . . . . . . 10 $/\$ $/\$
15		qbtwnxr 11490	. . . . . . . . . 10 $/\$ $/\$ $/\$
16	9, 13, 14, 15	syl3anc 1267	. . . . . . . . 9 $/\$ $/\$ $/\$
17		reeanv 2957	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		df-ov 6291	. . . . . . . . . . . . . 14
19		opelxpi 4865	. . . . . . . . . . . . . . . 16 $/\$
20	19	3ad2ant2 1029	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		ffun 5729	. . . . . . . . . . . . . . . . 17
22	3, 21	ax-mp 5	. . . . . . . . . . . . . . . 16
23		qssre 11271	. . . . . . . . . . . . . . . . . . 19
24		ressxr 9681	. . . . . . . . . . . . . . . . . . 19
25	23, 24	sstri 3440	. . . . . . . . . . . . . . . . . 18
26		xpss12 4939	. . . . . . . . . . . . . . . . . 18 $/\$
27	25, 25, 26	mp2an 677	. . . . . . . . . . . . . . . . 17
28	3	fdmi 5732	. . . . . . . . . . . . . . . . 17
29	27, 28	sseqtr4i 3464	. . . . . . . . . . . . . . . 16
30		funfvima2 6139	. . . . . . . . . . . . . . . 16 $/\$
31	22, 29, 30	mp2an 677	. . . . . . . . . . . . . . 15
32	20, 31	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	18, 32	syl5eqel 2532	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	9	3ad2ant1 1028	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simp3lr 1079	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simp3rl 1080	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp2l 1033	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	25, 37	sseldi 3429	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		simp2r 1034	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	25, 39	sseldi 3429	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		elioo1 11673	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
42	38, 40, 41	syl2anc 666	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	34, 35, 36, 42	mpbir3and 1190	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	6	3ad2ant1 1028	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp3ll 1078	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46		xrltle 11445	. . . . . . . . . . . . . . . . 17 $/\$
47	44, 38, 46	syl2anc 666	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	45, 47	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		iooss1 11668	. . . . . . . . . . . . . . 15 $/\$
50	44, 48, 49	syl2anc 666	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	13	3ad2ant1 1028	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simp3rr 1081	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		xrltle 11445	. . . . . . . . . . . . . . . . 17 $/\$
54	40, 51, 53	syl2anc 666	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 54	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		iooss2 11669	. . . . . . . . . . . . . . 15 $/\$
57	51, 55, 56	syl2anc 666	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	50, 57	sstrd 3441	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59		eleq2 2517	. . . . . . . . . . . . . . 15
60		sseq1 3452	. . . . . . . . . . . . . . 15
61	59, 60	anbi12d 716	. . . . . . . . . . . . . 14 $/\$ $/\$
62	61	rspcev 3149	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	33, 43, 58, 62	syl12anc 1265	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3exp 1206	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	64	rexlimdvv 2884	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	17, 65	syl5bir 222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	12, 16, 66	mp2and 684	. . . . . . . 8 $/\$ $/\$ $/\$
68	67	ralrimiva 2801	. . . . . . 7 $/\$ $/\$
69		qtopbas 21773	. . . . . . . 8
70		eltg2b 19967	. . . . . . . 8 $/\$
71	69, 70	ax-mp 5	. . . . . . 7 $/\$
72	68, 71	sylibr 216	. . . . . 6 $/\$
73	72	rgen2a 2814	. . . . 5
74		ffnov 6397	. . . . 5 $/\$
75	5, 73, 74	mpbir2an 930	. . . 4
76		frn 5733	. . . 4
77	75, 76	ax-mp 5	. . 3
78		2basgen 19999	. . 3 $/\$
79	2, 77, 78	mp2an 677	. 2
80	1, 79	eqtr2i 2473	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 188 $/\$ wa 371 $/\$ w3a 984

wceq 1443

wcel 1886

wral 2736

wrex 2737

wss 3403

cpw 3950

cop 3973 class class class wbr 4401

cxp 4831

cdm 4833

crn 4834

cima 4836

wfun 5575

wfn 5576

wf 5577

cfv 5581 (class class class)co 6288

cr 9535

cxr 9671

clt 9672

cle 9673

cq 11261

cioo 11632

ctg 15329

ctb 19913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-sup 7953 df-inf 7954 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-n0 10867 df-z 10935 df-uz 11157 df-q 11262 df-ioo 11636 df-topgen 15335 df-bases 19915

This theorem is referenced by: re2ndc 21812 opnmblALT 22554 mbfimaopnlem 22604

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