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Theorem metss 12702

Description: Two ways of saying that metric

generates a finer topology than metric

. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

**Hypotheses**
Ref	Expression
metequiv.3
metequiv.4

**Assertion**
Ref	Expression
metss	$/\$

Distinct variable groups:

Proof of Theorem metss Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metequiv.3	. . . . 5
2	1	mopnval 12650	. . . 4
3	2	adantr 274	. . 3 $/\$
4		metequiv.4	. . . . 5
5	4	mopnval 12650	. . . 4
6	5	adantl 275	. . 3 $/\$
7	3, 6	sseq12d 3133	. 2 $/\$
8		blbas 12641	. . . 4
9	8	adantr 274	. . 3 $/\$
10		unirnbl 12631	. . . . 5
11	10	adantr 274	. . . 4 $/\$
12		unirnbl 12631	. . . . 5
13	12	adantl 275	. . . 4 $/\$
14	11, 13	eqtr4d 2176	. . 3 $/\$
15		tgss2 12287	. . 3 $/\$ $/\$
16	9, 14, 15	syl2anc 409	. 2 $/\$ $/\$
17	11	raleqdv 2635	. . 3 $/\$ $/\$ $/\$
18		blssex 12638	. . . . . . . 8 $/\$ $/\$
19	18	adantll 468	. . . . . . 7 $/\$ $/\$ $/\$
20	19	imbi2d 229	. . . . . 6 $/\$ $/\$ $/\$
21	20	ralbidv 2438	. . . . 5 $/\$ $/\$ $/\$
22		rpxr 9478	. . . . . . . . . . 11
23		blelrn 12628	. . . . . . . . . . 11 $/\$ $/\$
24	22, 23	syl3an3 1252	. . . . . . . . . 10 $/\$ $/\$
25		blcntr 12624	. . . . . . . . . 10 $/\$ $/\$
26		eleq2 2204	. . . . . . . . . . . . 13
27		sseq2 3126	. . . . . . . . . . . . . 14
28	27	rexbidv 2439	. . . . . . . . . . . . 13
29	26, 28	imbi12d 233	. . . . . . . . . . . 12
30	29	rspcv 2789	. . . . . . . . . . 11
31	30	com23 78	. . . . . . . . . 10
32	24, 25, 31	sylc 62	. . . . . . . . 9 $/\$ $/\$
33	32	3expa 1182	. . . . . . . 8 $/\$ $/\$
34	33	adantllr 473	. . . . . . 7 $/\$ $/\$ $/\$
35	34	ralrimdva 2515	. . . . . 6 $/\$ $/\$
36		blss 12636	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	3expb 1183	. . . . . . . . . . . 12 $/\$ $/\$
38	37	adantlr 469	. . . . . . . . . . 11 $/\$ $/\$ $/\$
39	38	adantlr 469	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
40		r19.29 2572	. . . . . . . . . . . 12 $/\$ $/\$
41		sstr 3110	. . . . . . . . . . . . . . . 16 $/\$
42	41	expcom 115	. . . . . . . . . . . . . . 15
43	42	reximdv 2536	. . . . . . . . . . . . . 14
44	43	impcom 124	. . . . . . . . . . . . 13 $/\$
45	44	rexlimivw 2548	. . . . . . . . . . . 12 $/\$
46	40, 45	syl 14	. . . . . . . . . . 11 $/\$
47	46	ex 114	. . . . . . . . . 10
48	39, 47	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
49	48	expr 373	. . . . . . . 8 $/\$ $/\$ $/\$
50	49	com23 78	. . . . . . 7 $/\$ $/\$ $/\$
51	50	ralrimdva 2515	. . . . . 6 $/\$ $/\$
52	35, 51	impbid 128	. . . . 5 $/\$ $/\$
53	21, 52	bitrd 187	. . . 4 $/\$ $/\$ $/\$
54	53	ralbidva 2434	. . 3 $/\$ $/\$
55	17, 54	bitrd 187	. 2 $/\$ $/\$
56	7, 16, 55	3bitrd 213	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 963

wceq 1332

wcel 1481

wral 2417

wrex 2418

wss 3076

cuni 3744

crn 4548

cfv 5131 (class class class)co 5782

cxr 7823

crp 9470

ctg 12174

cxmet 12188

cbl 12190

cmopn 12193

ctb 12248

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-map 6552 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-bl 12198 df-mopn 12199 df-top 12204 df-bases 12249

This theorem is referenced by: metequiv 12703 metss2 12706

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