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Mirrors > Home > MPE Home > Th. List > mopnfss | Structured version Visualization version GIF version |
Description: The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
mopnfss | β’ (π· β (βMetβπ) β π½ β π« π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4941 | . 2 β’ π½ β π« βͺ π½ | |
2 | mopnval.1 | . . . 4 β’ π½ = (MetOpenβπ·) | |
3 | 2 | mopnuni 23873 | . . 3 β’ (π· β (βMetβπ) β π = βͺ π½) |
4 | 3 | pweqd 4612 | . 2 β’ (π· β (βMetβπ) β π« π = π« βͺ π½) |
5 | 1, 4 | sseqtrrid 4030 | 1 β’ (π· β (βMetβπ) β π½ β π« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wss 3943 π« cpw 4595 βͺ cuni 4900 βcfv 6531 βMetcxmet 20860 MetOpencmopn 20865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 ax-pre-sup 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-1st 7956 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9418 df-inf 9419 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-div 11853 df-nn 12194 df-2 12256 df-n0 12454 df-z 12540 df-uz 12804 df-q 12914 df-rp 12956 df-xneg 13073 df-xadd 13074 df-xmul 13075 df-topgen 17370 df-psmet 20867 df-xmet 20868 df-bl 20870 df-mopn 20871 df-top 22322 df-topon 22339 df-bases 22375 |
This theorem is referenced by: (None) |
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