![]() |
Mathbox for ML |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > topdifinffin | Structured version Visualization version GIF version |
Description: Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets π₯ of π΄ such that the complement of π₯ in π΄ is infinite, or π₯ is the empty set, or π₯ is all of π΄, is a topology only if π΄ is finite. (Contributed by ML, 17-Jul-2020.) |
Ref | Expression |
---|---|
topdifinf.t | β’ π = {π₯ β π« π΄ β£ (Β¬ (π΄ β π₯) β Fin β¨ (π₯ = β β¨ π₯ = π΄))} |
Ref | Expression |
---|---|
topdifinffin | β’ (π β (TopOnβπ΄) β π΄ β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topdifinf.t | . . 3 β’ π = {π₯ β π« π΄ β£ (Β¬ (π΄ β π₯) β Fin β¨ (π₯ = β β¨ π₯ = π΄))} | |
2 | difeq2 4112 | . . . . . . 7 β’ (π₯ = π¦ β (π΄ β π₯) = (π΄ β π¦)) | |
3 | 2 | eleq1d 2813 | . . . . . 6 β’ (π₯ = π¦ β ((π΄ β π₯) β Fin β (π΄ β π¦) β Fin)) |
4 | 3 | notbid 318 | . . . . 5 β’ (π₯ = π¦ β (Β¬ (π΄ β π₯) β Fin β Β¬ (π΄ β π¦) β Fin)) |
5 | eqeq1 2731 | . . . . . 6 β’ (π₯ = π¦ β (π₯ = β β π¦ = β )) | |
6 | eqeq1 2731 | . . . . . 6 β’ (π₯ = π¦ β (π₯ = π΄ β π¦ = π΄)) | |
7 | 5, 6 | orbi12d 917 | . . . . 5 β’ (π₯ = π¦ β ((π₯ = β β¨ π₯ = π΄) β (π¦ = β β¨ π¦ = π΄))) |
8 | 4, 7 | orbi12d 917 | . . . 4 β’ (π₯ = π¦ β ((Β¬ (π΄ β π₯) β Fin β¨ (π₯ = β β¨ π₯ = π΄)) β (Β¬ (π΄ β π¦) β Fin β¨ (π¦ = β β¨ π¦ = π΄)))) |
9 | 8 | cbvrabv 3437 | . . 3 β’ {π₯ β π« π΄ β£ (Β¬ (π΄ β π₯) β Fin β¨ (π₯ = β β¨ π₯ = π΄))} = {π¦ β π« π΄ β£ (Β¬ (π΄ β π¦) β Fin β¨ (π¦ = β β¨ π¦ = π΄))} |
10 | 1, 9 | eqtri 2755 | . 2 β’ π = {π¦ β π« π΄ β£ (Β¬ (π΄ β π¦) β Fin β¨ (π¦ = β β¨ π¦ = π΄))} |
11 | 10 | topdifinffinlem 36749 | 1 β’ (π β (TopOnβπ΄) β π΄ β Fin) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β¨ wo 846 = wceq 1534 β wcel 2099 {crab 3427 β cdif 3941 β c0 4318 π« cpw 4598 βcfv 6542 Fincfn 8953 TopOnctopon 22786 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-en 8954 df-fin 8957 df-topgen 17410 df-top 22770 df-topon 22787 |
This theorem is referenced by: topdifinf 36751 |
Copyright terms: Public domain | W3C validator |