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Mirrors > Home > MPE Home > Th. List > 0nnei | Structured version Visualization version GIF version |
Description: The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.) |
Ref | Expression |
---|---|
0nnei | β’ ((π½ β Top β§ π β β ) β Β¬ β β ((neiβπ½)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnei 23013 | . . . . 5 β’ ((π½ β Top β§ β β ((neiβπ½)βπ)) β π β β ) | |
2 | ss0b 4398 | . . . . 5 β’ (π β β β π = β ) | |
3 | 1, 2 | sylib 217 | . . . 4 β’ ((π½ β Top β§ β β ((neiβπ½)βπ)) β π = β ) |
4 | 3 | ex 412 | . . 3 β’ (π½ β Top β (β β ((neiβπ½)βπ) β π = β )) |
5 | 4 | necon3ad 2950 | . 2 β’ (π½ β Top β (π β β β Β¬ β β ((neiβπ½)βπ))) |
6 | 5 | imp 406 | 1 β’ ((π½ β Top β§ π β β ) β Β¬ β β ((neiβπ½)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 β wss 3947 β c0 4323 βcfv 6548 Topctop 22794 neicnei 23000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-top 22795 df-nei 23001 |
This theorem is referenced by: neifil 23783 |
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