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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnneieqvv | Structured version Visualization version GIF version |
Description: The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 46698 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.) |
Ref | Expression |
---|---|
opnneir.1 | β’ (π β π½ β Top) |
opnneilv.2 | β’ ((π β§ π¦ β π₯) β (π β π)) |
opnneil.3 | β’ ((π β§ π₯ = π¦) β (π β π)) |
Ref | Expression |
---|---|
opnneieqvv | β’ (π β (βπ₯ β ((neiβπ½)βπ)π β βπ¦ β π½ (π β π¦ β§ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opnneir.1 | . . 3 β’ (π β π½ β Top) | |
2 | opnneilv.2 | . . 3 β’ ((π β§ π¦ β π₯) β (π β π)) | |
3 | opnneil.3 | . . 3 β’ ((π β§ π₯ = π¦) β (π β π)) | |
4 | 1, 2, 3 | opnneieqv 46698 | . 2 β’ (π β (βπ₯ β ((neiβπ½)βπ)π β βπ₯ β π½ (π β π₯ β§ π))) |
5 | 3 | opnneilem 46693 | . 2 β’ (π β (βπ₯ β π½ (π β π₯ β§ π) β βπ¦ β π½ (π β π¦ β§ π))) |
6 | 4, 5 | bitrd 279 | 1 β’ (π β (βπ₯ β ((neiβπ½)βπ)π β βπ¦ β π½ (π β π¦ β§ π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β wcel 2107 βwrex 3072 β wss 3909 βcfv 6492 Topctop 22165 neicnei 22371 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-top 22166 df-nei 22372 |
This theorem is referenced by: (None) |
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