blbas - Intuitionistic Logic Explorer

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Theorem blbas 14204

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 14203	. . . . . 6 $/\$ $/\$ $/\$
2		simpll 527	. . . . . . 7 $/\$ $/\$ $/\$
3		elinel1 3333	. . . . . . . . . 10
4		elunii 3826	. . . . . . . . . 10 $/\$
5	3, 4	sylan 283	. . . . . . . . 9 $/\$
6	5	ad2ant2lr 510	. . . . . . . 8 $/\$ $/\$ $/\$
7		unirnbl 14194	. . . . . . . . 9
8	7	ad2antrr 488	. . . . . . . 8 $/\$ $/\$ $/\$
9	6, 8	eleqtrd 2266	. . . . . . 7 $/\$ $/\$ $/\$
10		blssex 14201	. . . . . . 7 $/\$ $/\$
11	2, 9, 10	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$
12	1, 11	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
13	12	ex 115	. . . 4 $/\$ $/\$ $/\$
14	13	ralrimdva 2567	. . 3 $/\$ $/\$
15	14	ralrimivv 2568	. 2 $/\$
16		blex 14158	. . 3
17		rnexg 4904	. . 3
18		isbasis2g 13816	. . 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 1363

wcel 2158

wral 2465

wrex 2466

cvv 2749

cin 3140

wss 3141

cuni 3821

crn 4639

cfv 5228 (class class class)co 5888

crp 9666

cxmet 13697

cbl 13699

ctb 13813

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-map 6663 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-psmet 13704 df-xmet 13705 df-bl 13707 df-bases 13814

This theorem is referenced by: mopnval 14213 mopntopon 14214 elmopn 14217 blssopn 14256 metss 14265