re2ndc - Metamath Proof Explorer

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Theorem re2ndc 22604

Description: The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

**Assertion**
Ref	Expression
re2ndc

**Proof of Theorem re2ndc**
Step	Hyp	Ref	Expression
1		eqid 2622	. . 3
2	1	tgqioo 22603	. 2
3		qtopbas 22563	. . 3
4		omelon 8543	. . . . . 6
5		qnnen 14942	. . . . . . . . 9
6		xpen 8123	. . . . . . . . 9 $/\$
7	5, 5, 6	mp2an 708	. . . . . . . 8
8		xpnnen 14939	. . . . . . . 8
9	7, 8	entri 8010	. . . . . . 7
10		nnenom 12779	. . . . . . 7
11	9, 10	entr2i 8011	. . . . . 6
12		isnumi 8772	. . . . . 6 $/\$
13	4, 11, 12	mp2an 708	. . . . 5
14		ioof 12271	. . . . . . 7
15		ffun 6048	. . . . . . 7
16	14, 15	ax-mp 5	. . . . . 6
17		qssre 11798	. . . . . . . . 9
18		ressxr 10083	. . . . . . . . 9
19	17, 18	sstri 3612	. . . . . . . 8
20		xpss12 5225	. . . . . . . 8 $/\$
21	19, 19, 20	mp2an 708	. . . . . . 7
22	14	fdmi 6052	. . . . . . 7
23	21, 22	sseqtr4i 3638	. . . . . 6
24		fores 6124	. . . . . 6 $/\$
25	16, 23, 24	mp2an 708	. . . . 5
26		fodomnum 8880	. . . . 5
27	13, 25, 26	mp2 9	. . . 4
28	9, 10	entri 8010	. . . 4
29		domentr 8015	. . . 4 $/\$
30	27, 28, 29	mp2an 708	. . 3
31		2ndci 21251	. . 3 $/\$
32	3, 30, 31	mp2an 708	. 2
33	2, 32	eqeltri 2697	1

Colors of variables: wff setvar class

Syntax hints:

wcel 1990

wss 3574

cpw 4158 class class class wbr 4653

cxp 5112

cdm 5114

crn 5115

cres 5116

cima 5117

con0 5723

wfun 5882

wf 5884

wfo 5886

cfv 5888

com 7065

cen 7952

cdom 7953

ccrd 8761

cr 9935

cxr 10073

cn 11020

cq 11788

cioo 12175

ctg 16098

ctb 20749

c2ndc 21241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ioo 12179 df-topgen 16104 df-bases 20750 df-2ndc 21243

This theorem is referenced by: (None)