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Theorem blss 22230

Description: Any point

in a ball

can be centered in another ball that is a subset of

. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)

**Assertion**
Ref	Expression
blss	$/\$ $/\$

Distinct variable groups:

Proof of Theorem blss Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blrn 22214	. . 3
2		elbl 22193	. . . . . . 7 $/\$ $/\$ $/\$
3		simpl1 1064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
4		simpl2 1065	. . . . . . . . . . 11 $/\$ $/\$ $/\$
5		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$
6		xmetcl 22136	. . . . . . . . . . 11 $/\$ $/\$
7	3, 4, 5, 6	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
8		simpl3 1066	. . . . . . . . . 10 $/\$ $/\$ $/\$
9		qbtwnxr 12031	. . . . . . . . . . 11 $/\$ $/\$ $/\$
10	9	3expia 1267	. . . . . . . . . 10 $/\$ $/\$
11	7, 8, 10	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
12		qre 11793	. . . . . . . . . . 11
13		simpll1 1100	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplr 792	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpll2 1101	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		xmetsym 22152	. . . . . . . . . . . . . . . 16 $/\$ $/\$
17	13, 14, 15, 16	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simprrl 804	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	17, 18	eqbrtrd 4675	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 794	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		xmetcl 22136	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
22	13, 14, 15, 21	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		rexr 10085	. . . . . . . . . . . . . . . . . . 19
24	23	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		xrltle 11982	. . . . . . . . . . . . . . . . . 18 $/\$
26	22, 24, 25	syl2anc 693	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	19, 26	mpd 15	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		xmetlecl 22151	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
29	13, 14, 15, 20, 27, 28	syl122anc 1335	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		difrp 11868	. . . . . . . . . . . . . . 15 $/\$
31	29, 20, 30	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	19, 31	mpbid 222	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	20, 29	resubcld 10458	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		xrleid 11983	. . . . . . . . . . . . . . . . 17
35	22, 34	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	20	recnd 10068	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	29	recnd 10068	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	36, 37	nncand 10397	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	35, 38	breqtrrd 4681	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		blss2 22209	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
41	13, 14, 15, 33, 20, 39, 40	syl33anc 1341	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simpll3 1102	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simprrr 805	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		xrltle 11982	. . . . . . . . . . . . . . . . 17 $/\$
45	24, 42, 44	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43, 45	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		ssbl 22228	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
48	13, 15, 24, 42, 46, 47	syl221anc 1337	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	41, 48	sstrd 3613	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		oveq2 6658	. . . . . . . . . . . . . . 15
51	50	sseq1d 3632	. . . . . . . . . . . . . 14
52	51	rspcev 3309	. . . . . . . . . . . . 13 $/\$
53	32, 49, 52	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	53	expr 643	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
55	12, 54	sylan2 491	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56	55	rexlimdva 3031	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
57	11, 56	syld 47	. . . . . . . 8 $/\$ $/\$ $/\$
58	57	expimpd 629	. . . . . . 7 $/\$ $/\$ $/\$
59	2, 58	sylbid 230	. . . . . 6 $/\$ $/\$
60		eleq2 2690	. . . . . . 7
61		sseq2 3627	. . . . . . . 8
62	61	rexbidv 3052	. . . . . . 7
63	60, 62	imbi12d 334	. . . . . 6
64	59, 63	syl5ibrcom 237	. . . . 5 $/\$ $/\$
65	64	3expib 1268	. . . 4 $/\$
66	65	rexlimdvv 3037	. . 3
67	1, 66	sylbid 230	. 2
68	67	3imp 1256	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

wss 3574 class class class wbr 4653

crn 5115

cfv 5888 (class class class)co 6650

cr 9935

cxr 10073

clt 10074

cle 10075

cmin 10266

cq 11788

crp 11832

cxmt 19731

cbl 19733

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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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-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-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-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-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-psmet 19738 df-xmet 19739 df-bl 19741

This theorem is referenced by: blssex 22232 blin2 22234 metss 22313 metcnp3 22345

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