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Theorem metss 12422

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 12370	. . . 4
3	2	adantr 272	. . 3 $/\$
4		metequiv.4	. . . . 5
5	4	mopnval 12370	. . . 4
6	5	adantl 273	. . 3 $/\$
7	3, 6	sseq12d 3078	. 2 $/\$
8		blbas 12361	. . . 4
9	8	adantr 272	. . 3 $/\$
10		unirnbl 12351	. . . . 5
11	10	adantr 272	. . . 4 $/\$
12		unirnbl 12351	. . . . 5
13	12	adantl 273	. . . 4 $/\$
14	11, 13	eqtr4d 2135	. . 3 $/\$
15		tgss2 12030	. . 3 $/\$ $/\$
16	9, 14, 15	syl2anc 406	. 2 $/\$ $/\$
17	11	raleqdv 2590	. . 3 $/\$ $/\$ $/\$
18		blssex 12358	. . . . . . . 8 $/\$ $/\$
19	18	adantll 463	. . . . . . 7 $/\$ $/\$ $/\$
20	19	imbi2d 229	. . . . . 6 $/\$ $/\$ $/\$
21	20	ralbidv 2396	. . . . 5 $/\$ $/\$ $/\$
22		rpxr 9298	. . . . . . . . . . 11
23		blelrn 12348	. . . . . . . . . . 11 $/\$ $/\$
24	22, 23	syl3an3 1219	. . . . . . . . . 10 $/\$ $/\$
25		blcntr 12344	. . . . . . . . . 10 $/\$ $/\$
26		eleq2 2163	. . . . . . . . . . . . 13
27		sseq2 3071	. . . . . . . . . . . . . 14
28	27	rexbidv 2397	. . . . . . . . . . . . 13
29	26, 28	imbi12d 233	. . . . . . . . . . . 12
30	29	rspcv 2740	. . . . . . . . . . 11
31	30	com23 78	. . . . . . . . . 10
32	24, 25, 31	sylc 62	. . . . . . . . 9 $/\$ $/\$
33	32	3expa 1149	. . . . . . . 8 $/\$ $/\$
34	33	adantllr 468	. . . . . . 7 $/\$ $/\$ $/\$
35	34	ralrimdva 2471	. . . . . 6 $/\$ $/\$
36		blss 12356	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	3expb 1150	. . . . . . . . . . . 12 $/\$ $/\$
38	37	adantlr 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
39	38	adantlr 464	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
40		r19.29 2528	. . . . . . . . . . . 12 $/\$ $/\$
41		sstr 3055	. . . . . . . . . . . . . . . 16 $/\$
42	41	expcom 115	. . . . . . . . . . . . . . 15
43	42	reximdv 2492	. . . . . . . . . . . . . 14
44	43	impcom 124	. . . . . . . . . . . . 13 $/\$
45	44	rexlimivw 2504	. . . . . . . . . . . 12 $/\$
46	40, 45	syl 14	. . . . . . . . . . 11 $/\$
47	46	ex 114	. . . . . . . . . 10
48	39, 47	syl5com 29	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
49	48	expr 370	. . . . . . . 8 $/\$ $/\$ $/\$
50	49	com23 78	. . . . . . 7 $/\$ $/\$ $/\$
51	50	ralrimdva 2471	. . . . . 6 $/\$ $/\$
52	35, 51	impbid 128	. . . . 5 $/\$ $/\$
53	21, 52	bitrd 187	. . . 4 $/\$ $/\$ $/\$
54	53	ralbidva 2392	. . 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 930

wceq 1299

wcel 1448

wral 2375

wrex 2376

wss 3021

cuni 3683

crn 4478

cfv 5059 (class class class)co 5706

cxr 7671

crp 9291

ctg 11917

cxmet 11931

cbl 11933

cmopn 11936

ctb 11991

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 ax-arch 7614 ax-caucvg 7615

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-isom 5068 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-map 6474 df-sup 6786 df-inf 6787 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-n0 8830 df-z 8907 df-uz 9177 df-q 9262 df-rp 9292 df-xneg 9400 df-xadd 9401 df-seqfrec 10060 df-exp 10134 df-cj 10455 df-re 10456 df-im 10457 df-rsqrt 10610 df-abs 10611 df-topgen 11923 df-psmet 11938 df-xmet 11939 df-bl 11941 df-mopn 11942 df-top 11947 df-bases 11992

This theorem is referenced by: metequiv 12423 metss2 12426

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