opnreen - Metamath Proof Explorer

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Theorem opnreen 22634

Description: Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)

**Assertion**
Ref	Expression
opnreen	$/\$

Proof of Theorem opnreen Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10027	. . . . 5
2		elssuni 4467	. . . . . 6
3		uniretop 22566	. . . . . 6
4	2, 3	syl6sseqr 3652	. . . . 5
5		ssdomg 8001	. . . . 5
6	1, 4, 5	mpsyl 68	. . . 4
7		rpnnen 14956	. . . 4
8		domentr 8015	. . . 4 $/\$
9	6, 7, 8	sylancl 694	. . 3
10	9	adantr 481	. 2 $/\$
11		n0 3931	. . . 4
12	4	sselda 3603	. . . . . . . . . 10 $/\$
13		rpnnen2 14955	. . . . . . . . . . . . 13
14		rphalfcl 11858	. . . . . . . . . . . . . . . 16
15	14	rpred 11872	. . . . . . . . . . . . . . 15
16		resubcl 10345	. . . . . . . . . . . . . . 15 $/\$
17	15, 16	sylan2 491	. . . . . . . . . . . . . 14 $/\$
18		readdcl 10019	. . . . . . . . . . . . . . 15 $/\$
19	15, 18	sylan2 491	. . . . . . . . . . . . . 14 $/\$
20		simpl 473	. . . . . . . . . . . . . . 15 $/\$
21		ltsubrp 11866	. . . . . . . . . . . . . . . 16 $/\$
22	14, 21	sylan2 491	. . . . . . . . . . . . . . 15 $/\$
23		ltaddrp 11867	. . . . . . . . . . . . . . . 16 $/\$
24	14, 23	sylan2 491	. . . . . . . . . . . . . . 15 $/\$
25	17, 20, 19, 22, 24	lttrd 10198	. . . . . . . . . . . . . 14 $/\$
26		iccen 12317	. . . . . . . . . . . . . 14 $/\$ $/\$
27	17, 19, 25, 26	syl3anc 1326	. . . . . . . . . . . . 13 $/\$
28		domentr 8015	. . . . . . . . . . . . 13 $/\$
29	13, 27, 28	sylancr 695	. . . . . . . . . . . 12 $/\$
30		ovex 6678	. . . . . . . . . . . . 13
31		rpre 11839	. . . . . . . . . . . . . . . 16
32		resubcl 10345	. . . . . . . . . . . . . . . 16 $/\$
33	31, 32	sylan2 491	. . . . . . . . . . . . . . 15 $/\$
34	33	rexrd 10089	. . . . . . . . . . . . . 14 $/\$
35		readdcl 10019	. . . . . . . . . . . . . . . 16 $/\$
36	31, 35	sylan2 491	. . . . . . . . . . . . . . 15 $/\$
37	36	rexrd 10089	. . . . . . . . . . . . . 14 $/\$
38	20	recnd 10068	. . . . . . . . . . . . . . . . 17 $/\$
39	15	adantl 482	. . . . . . . . . . . . . . . . . 18 $/\$
40	39	recnd 10068	. . . . . . . . . . . . . . . . 17 $/\$
41	38, 40, 40	subsub4d 10423	. . . . . . . . . . . . . . . 16 $/\$
42	31	adantl 482	. . . . . . . . . . . . . . . . . . 19 $/\$
43	42	recnd 10068	. . . . . . . . . . . . . . . . . 18 $/\$
44	43	2halvesd 11278	. . . . . . . . . . . . . . . . 17 $/\$
45	44	oveq2d 6666	. . . . . . . . . . . . . . . 16 $/\$
46	41, 45	eqtrd 2656	. . . . . . . . . . . . . . 15 $/\$
47	14	adantl 482	. . . . . . . . . . . . . . . 16 $/\$
48	17, 47	ltsubrpd 11904	. . . . . . . . . . . . . . 15 $/\$
49	46, 48	eqbrtrrd 4677	. . . . . . . . . . . . . 14 $/\$
50		ltaddrp 11867	. . . . . . . . . . . . . . . 16 $/\$
51	19, 47, 50	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$
52	38, 40, 40	addassd 10062	. . . . . . . . . . . . . . . 16 $/\$
53	44	oveq2d 6666	. . . . . . . . . . . . . . . 16 $/\$
54	52, 53	eqtrd 2656	. . . . . . . . . . . . . . 15 $/\$
55	51, 54	breqtrd 4679	. . . . . . . . . . . . . 14 $/\$
56		iccssioo 12242	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
57	34, 37, 49, 55, 56	syl22anc 1327	. . . . . . . . . . . . 13 $/\$
58		ssdomg 8001	. . . . . . . . . . . . 13
59	30, 57, 58	mpsyl 68	. . . . . . . . . . . 12 $/\$
60		domtr 8009	. . . . . . . . . . . 12 $/\$
61	29, 59, 60	syl2anc 693	. . . . . . . . . . 11 $/\$
62		eqid 2622	. . . . . . . . . . . . 13
63	62	bl2ioo 22595	. . . . . . . . . . . 12 $/\$
64	31, 63	sylan2 491	. . . . . . . . . . 11 $/\$
65	61, 64	breqtrrd 4681	. . . . . . . . . 10 $/\$
66	12, 65	sylan 488	. . . . . . . . 9 $/\$ $/\$
67	66	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$
68		simplll 798	. . . . . . . . 9 $/\$ $/\$ $/\$
69		simpr 477	. . . . . . . . 9 $/\$ $/\$ $/\$
70		ssdomg 8001	. . . . . . . . 9
71	68, 69, 70	sylc 65	. . . . . . . 8 $/\$ $/\$ $/\$
72		domtr 8009	. . . . . . . 8 $/\$
73	67, 71, 72	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
74		eqid 2622	. . . . . . . . . 10
75	62, 74	tgioo 22599	. . . . . . . . 9
76	75	eleq2i 2693	. . . . . . . 8
77	62	rexmet 22594	. . . . . . . . 9
78	74	mopni2 22298	. . . . . . . . 9 $/\$ $/\$
79	77, 78	mp3an1 1411	. . . . . . . 8 $/\$
80	76, 79	sylanb 489	. . . . . . 7 $/\$
81	73, 80	r19.29a 3078	. . . . . 6 $/\$
82	81	ex 450	. . . . 5
83	82	exlimdv 1861	. . . 4
84	11, 83	syl5bi 232	. . 3
85	84	imp 445	. 2 $/\$
86		sbth 8080	. 2 $/\$
87	10, 85, 86	syl2anc 693	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wex 1704

wcel 1990

wne 2794

wrex 2913

cvv 3200

wss 3574

c0 3915

cpw 4158

cuni 4436 class class class wbr 4653

cxp 5112

crn 5115

cres 5116

ccom 5118

cfv 5888 (class class class)co 6650

cen 7952

cdom 7953

cr 9935

cc0 9936

c1 9937

caddc 9939

cxr 10073

clt 10074

cmin 10266

cdiv 10684

cn 11020

c2 11070

crp 11832

cioo 12175

cicc 12178

cabs 13974

ctg 16098

cxmt 19731

cbl 19733

cmopn 19736

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-fal 1489 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-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 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-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750

This theorem is referenced by: rectbntr0 22635

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