blbas - Intuitionistic Logic Explorer

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Theorem blbas 12605

Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)

**Assertion**
Ref	Expression
blbas

Proof of Theorem blbas Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blin2 12604	. . . . . 6 $/\$ $/\$ $/\$
2		simpll 518	. . . . . . 7 $/\$ $/\$ $/\$
3		elinel1 3262	. . . . . . . . . 10
4		elunii 3741	. . . . . . . . . 10 $/\$
5	3, 4	sylan 281	. . . . . . . . 9 $/\$
6	5	ad2ant2lr 501	. . . . . . . 8 $/\$ $/\$ $/\$
7		unirnbl 12595	. . . . . . . . 9
8	7	ad2antrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
9	6, 8	eleqtrd 2218	. . . . . . 7 $/\$ $/\$ $/\$
10		blssex 12602	. . . . . . 7 $/\$ $/\$
11	2, 9, 10	syl2anc 408	. . . . . 6 $/\$ $/\$ $/\$ $/\$
12	1, 11	mpbird 166	. . . . 5 $/\$ $/\$ $/\$ $/\$
13	12	ex 114	. . . 4 $/\$ $/\$ $/\$
14	13	ralrimdva 2512	. . 3 $/\$ $/\$
15	14	ralrimivv 2513	. 2 $/\$
16		blex 12559	. . 3
17		rnexg 4804	. . 3
18		isbasis2g 12215	. . 3 $/\$
19	16, 17, 18	3syl 17	. 2 $/\$
20	15, 19	mpbird 166	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1331

wcel 1480

wral 2416

wrex 2417

cvv 2686

cin 3070

wss 3071

cuni 3736

crn 4540

cfv 5123 (class class class)co 5774

crp 9444

cxmet 12152

cbl 12154

ctb 12212

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-xneg 9562 df-xadd 9563 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-psmet 12159 df-xmet 12160 df-bl 12162 df-bases 12213

This theorem is referenced by: mopnval 12614 mopntopon 12615 elmopn 12618 blssopn 12657 metss 12666