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Mirrors > Home > MPE Home > Th. List > neisspw | Structured version Visualization version GIF version |
Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.) |
Ref | Expression |
---|---|
neifval.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
neisspw | β’ (π½ β Top β ((neiβπ½)βπ) β π« π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 | . . . . 5 β’ π = βͺ π½ | |
2 | 1 | neii1 23028 | . . . 4 β’ ((π½ β Top β§ π£ β ((neiβπ½)βπ)) β π£ β π) |
3 | velpw 4609 | . . . 4 β’ (π£ β π« π β π£ β π) | |
4 | 2, 3 | sylibr 233 | . . 3 β’ ((π½ β Top β§ π£ β ((neiβπ½)βπ)) β π£ β π« π) |
5 | 4 | ex 411 | . 2 β’ (π½ β Top β (π£ β ((neiβπ½)βπ) β π£ β π« π)) |
6 | 5 | ssrdv 3986 | 1 β’ (π½ β Top β ((neiβπ½)βπ) β π« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3947 π« cpw 4604 βͺ cuni 4910 βcfv 6551 Topctop 22813 neicnei 23019 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-top 22814 df-nei 23020 |
This theorem is referenced by: hausflim 23903 flimclslem 23906 fclsfnflim 23949 |
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