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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 4772 | . . . . . 6 β’ (π β π β {π} β π) | |
2 | gneispace.x | . . . . . . 7 β’ π = βͺ π½ | |
3 | 2 | tpnei 22495 | . . . . . 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 408 | . . . 4 β’ ((π½ β Top β§ π β π) β π β ((neiβπ½)β{π})) |
6 | 5 | ne0d 4299 | . . 3 β’ ((π½ β Top β§ π β π) β ((neiβπ½)β{π}) β β ) |
7 | elnei 22485 | . . . . 5 β’ ((π½ β Top β§ π β π β§ π β ((neiβπ½)β{π})) β π β π) | |
8 | 7 | 3expia 1122 | . . . 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 513 | . 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 397 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 β wss 3914 β c0 4286 {csn 4590 βͺ cuni 4869 βcfv 6500 Topctop 22265 neicnei 22471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-top 22266 df-nei 22472 |
This theorem is referenced by: (None) |
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