blbas - Intuitionistic Logic Explorer

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Theorem blbas 15227

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 15226	. . . . . 6 $/\$ $/\$ $/\$
2		simpll 527	. . . . . . 7 $/\$ $/\$ $/\$
3		elinel1 3395	. . . . . . . . . 10
4		elunii 3903	. . . . . . . . . 10 $/\$
5	3, 4	sylan 283	. . . . . . . . 9 $/\$
6	5	ad2ant2lr 510	. . . . . . . 8 $/\$ $/\$ $/\$
7		unirnbl 15217	. . . . . . . . 9
8	7	ad2antrr 488	. . . . . . . 8 $/\$ $/\$ $/\$
9	6, 8	eleqtrd 2310	. . . . . . 7 $/\$ $/\$ $/\$
10		blssex 15224	. . . . . . 7 $/\$ $/\$
11	2, 9, 10	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$
12	1, 11	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
13	12	ex 115	. . . 4 $/\$ $/\$ $/\$
14	13	ralrimdva 2613	. . 3 $/\$ $/\$
15	14	ralrimivv 2614	. 2 $/\$
16		blex 15181	. . 3
17		rnexg 5003	. . 3
18		isbasis2g 14839	. . 3 $/\$
19	16, 17, 18	3syl 17	. 2 $/\$
20	15, 19	mpbird 167	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2202

wral 2511

wrex 2512

cvv 2803

cin 3200

wss 3201

cuni 3898

crn 4732

cfv 5333 (class class class)co 6028

crp 9932

cxmet 14615

cbl 14617

ctb 14836

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-map 6862 df-sup 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-xneg 10051 df-xadd 10052 df-seqfrec 10756 df-exp 10847 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-psmet 14622 df-xmet 14623 df-bl 14625 df-bases 14837

This theorem is referenced by: mopnval 15236 mopntopon 15237 elmopn 15240 blssopn 15279 metss 15288