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Mirrors > Home > ILE Home > Th. List > unirnbl | Unicode version |
Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
unirnbl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf 12957 | . . . 4 | |
2 | 1 | frnd 5341 | . . 3 |
3 | sspwuni 3944 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | 1rp 9584 | . . . 4 | |
6 | blcntr 12963 | . . . 4 | |
7 | 5, 6 | mp3an3 1315 | . . 3 |
8 | rpxr 9588 | . . . . 5 | |
9 | 5, 8 | ax-mp 5 | . . . 4 |
10 | blelrn 12967 | . . . 4 | |
11 | 9, 10 | mp3an3 1315 | . . 3 |
12 | elunii 3788 | . . 3 | |
13 | 7, 11, 12 | syl2anc 409 | . 2 |
14 | 4, 13 | eqelssd 3156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wss 3111 cpw 3553 cuni 3783 cxp 4596 crn 4599 cfv 5182 (class class class)co 5836 c1 7745 cxr 7923 crp 9580 cxmet 12527 cbl 12529 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-0lt1 7850 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-rp 9581 df-psmet 12534 df-xmet 12535 df-bl 12537 |
This theorem is referenced by: blbas 12980 mopntopon 12990 elmopn 12993 metss 13041 xmettx 13057 |
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