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Mirrors > Home > MPE Home > Th. List > elmopn2 | Structured version Visualization version GIF version |
Description: A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
elmopn2 | β’ (π· β (βMetβπ) β (π΄ β π½ β (π΄ β π β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 β’ π½ = (MetOpenβπ·) | |
2 | 1 | elmopn 24368 | . 2 β’ (π· β (βMetβπ) β (π΄ β π½ β (π΄ β π β§ βπ₯ β π΄ βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄)))) |
3 | ssel2 3977 | . . . . . 6 β’ ((π΄ β π β§ π₯ β π΄) β π₯ β π) | |
4 | blssex 24353 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π₯ β π) β (βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄)) | |
5 | 3, 4 | sylan2 591 | . . . . 5 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π₯ β π΄)) β (βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄)) |
6 | 5 | anassrs 466 | . . . 4 β’ (((π· β (βMetβπ) β§ π΄ β π) β§ π₯ β π΄) β (βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄) β βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄)) |
7 | 6 | ralbidva 3173 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π) β (βπ₯ β π΄ βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄) β βπ₯ β π΄ βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄)) |
8 | 7 | pm5.32da 577 | . 2 β’ (π· β (βMetβπ) β ((π΄ β π β§ βπ₯ β π΄ βπ§ β ran (ballβπ·)(π₯ β π§ β§ π§ β π΄)) β (π΄ β π β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄))) |
9 | 2, 8 | bitrd 278 | 1 β’ (π· β (βMetβπ) β (π΄ β π½ β (π΄ β π β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβπ·)π¦) β π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 β wss 3949 ran crn 5683 βcfv 6553 (class class class)co 7426 β+crp 13014 βMetcxmet 21271 ballcbl 21273 MetOpencmopn 21276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-bl 21281 df-mopn 21282 df-bases 22869 |
This theorem is referenced by: metrest 24453 tgioo 24732 xrsmopn 24748 recld2 24750 tpr2rico 33546 dya2icoseg2 33931 opnrebl 35837 opnrebl2 35838 |
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