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Theorem metss 15217

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 15165	. . . 4
3	2	adantr 276	. . 3 $/\$
4		metequiv.4	. . . . 5
5	4	mopnval 15165	. . . 4
6	5	adantl 277	. . 3 $/\$
7	3, 6	sseq12d 3258	. 2 $/\$
8		blbas 15156	. . . 4
9	8	adantr 276	. . 3 $/\$
10		unirnbl 15146	. . . . 5
11	10	adantr 276	. . . 4 $/\$
12		unirnbl 15146	. . . . 5
13	12	adantl 277	. . . 4 $/\$
14	11, 13	eqtr4d 2267	. . 3 $/\$
15		tgss2 14802	. . 3 $/\$ $/\$
16	9, 14, 15	syl2anc 411	. 2 $/\$ $/\$
17	11	raleqdv 2736	. . 3 $/\$ $/\$ $/\$
18		blssex 15153	. . . . . . . 8 $/\$ $/\$
19	18	adantll 476	. . . . . . 7 $/\$ $/\$ $/\$
20	19	imbi2d 230	. . . . . 6 $/\$ $/\$ $/\$
21	20	ralbidv 2532	. . . . 5 $/\$ $/\$ $/\$
22		rpxr 9895	. . . . . . . . . . 11
23		blelrn 15143	. . . . . . . . . . 11 $/\$ $/\$
24	22, 23	syl3an3 1308	. . . . . . . . . 10 $/\$ $/\$
25		blcntr 15139	. . . . . . . . . 10 $/\$ $/\$
26		eleq2 2295	. . . . . . . . . . . . 13
27		sseq2 3251	. . . . . . . . . . . . . 14
28	27	rexbidv 2533	. . . . . . . . . . . . 13
29	26, 28	imbi12d 234	. . . . . . . . . . . 12
30	29	rspcv 2906	. . . . . . . . . . 11
31	30	com23 78	. . . . . . . . . 10
32	24, 25, 31	sylc 62	. . . . . . . . 9 $/\$ $/\$
33	32	3expa 1229	. . . . . . . 8 $/\$ $/\$
34	33	adantllr 481	. . . . . . 7 $/\$ $/\$ $/\$
35	34	ralrimdva 2612	. . . . . 6 $/\$ $/\$
36		blss 15151	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	3expb 1230	. . . . . . . . . . . 12 $/\$ $/\$
38	37	adantlr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$
39	38	adantlr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
40		r19.29 2670	. . . . . . . . . . . 12 $/\$ $/\$
41		sstr 3235	. . . . . . . . . . . . . . . 16 $/\$
42	41	expcom 116	. . . . . . . . . . . . . . 15
43	42	reximdv 2633	. . . . . . . . . . . . . 14
44	43	impcom 125	. . . . . . . . . . . . 13 $/\$
45	44	rexlimivw 2646	. . . . . . . . . . . 12 $/\$
46	40, 45	syl 14	. . . . . . . . . . 11 $/\$
47	46	ex 115	. . . . . . . . . 10
48	39, 47	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
49	48	expr 375	. . . . . . . 8 $/\$ $/\$ $/\$
50	49	com23 78	. . . . . . 7 $/\$ $/\$ $/\$
51	50	ralrimdva 2612	. . . . . 6 $/\$ $/\$
52	35, 51	impbid 129	. . . . 5 $/\$ $/\$
53	21, 52	bitrd 188	. . . 4 $/\$ $/\$ $/\$
54	53	ralbidva 2528	. . 3 $/\$ $/\$
55	17, 54	bitrd 188	. 2 $/\$ $/\$
56	7, 16, 55	3bitrd 214	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1004

wceq 1397

wcel 2202

wral 2510

wrex 2511

wss 3200

cuni 3893

crn 4726

cfv 5326 (class class class)co 6017

cxr 8212

crp 9887

ctg 13336

cxmet 14549

cbl 14551

cmopn 14554

ctb 14765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-map 6818 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-bl 14559 df-mopn 14560 df-top 14721 df-bases 14766

This theorem is referenced by: metequiv 15218 metss2 15221

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