![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > gneispa | Structured version Visualization version GIF version |
Description: Each point π of the neighborhood space has at least one neighborhood; each neighborhood of π contains π. Axiom A of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.) |
Ref | Expression |
---|---|
gneispace.x | β’ π = βͺ π½ |
Ref | Expression |
---|---|
gneispa | β’ (π½ β Top β βπ β π (((neiβπ½)β{π}) β β β§ βπ β ((neiβπ½)β{π})π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4806 | . . . . . 6 β’ (π β π β {π} β π) | |
2 | gneispace.x | . . . . . . 7 β’ π = βͺ π½ | |
3 | 2 | tpnei 22976 | . . . . . 6 β’ (π½ β Top β ({π} β π β π β ((neiβπ½)β{π}))) |
4 | 1, 3 | imbitrid 243 | . . . . 5 β’ (π½ β Top β (π β π β π β ((neiβπ½)β{π}))) |
5 | 4 | imp 406 | . . . 4 β’ ((π½ β Top β§ π β π) β π β ((neiβπ½)β{π})) |
6 | 5 | ne0d 4330 | . . 3 β’ ((π½ β Top β§ π β π) β ((neiβπ½)β{π}) β β ) |
7 | elnei 22966 | . . . . 5 β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β π β π) | |
8 | 7 | 3expia 1118 | . . . 4 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)β{π}) β π β π)) |
9 | 8 | ralrimiv 3139 | . . 3 β’ ((π½ β Top β§ π β π) β βπ β ((neiβπ½)β{π})π β π) |
10 | 6, 9 | jca 511 | . 2 β’ ((π½ β Top β§ π β π) β (((neiβπ½)β{π}) β β β§ βπ β ((neiβπ½)β{π})π β π)) |
11 | 10 | ralrimiva 3140 | 1 β’ (π½ β Top β βπ β π (((neiβπ½)β{π}) β β β§ βπ β ((neiβπ½)β{π})π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β wss 3943 β c0 4317 {csn 4623 βͺ cuni 4902 βcfv 6536 Topctop 22746 neicnei 22952 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-top 22747 df-nei 22953 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |