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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 4109 | . . . . . . 7 β’ (π₯ = π¦ β (π΄ β π₯) = (π΄ β π¦)) | |
3 | 2 | eleq1d 2810 | . . . . . 6 β’ (π₯ = π¦ β ((π΄ β π₯) β Fin β (π΄ β π¦) β Fin)) |
4 | 3 | notbid 317 | . . . . 5 β’ (π₯ = π¦ β (Β¬ (π΄ β π₯) β Fin β Β¬ (π΄ β π¦) β Fin)) |
5 | eqeq1 2729 | . . . . . 6 β’ (π₯ = π¦ β (π₯ = β β π¦ = β )) | |
6 | eqeq1 2729 | . . . . . 6 β’ (π₯ = π¦ β (π₯ = π΄ β π¦ = π΄)) | |
7 | 5, 6 | orbi12d 916 | . . . . 5 β’ (π₯ = π¦ β ((π₯ = β β¨ π₯ = π΄) β (π¦ = β β¨ π¦ = π΄))) |
8 | 4, 7 | orbi12d 916 | . . . 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 3430 | . . 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 2753 | . 2 β’ π = {π¦ β π« π΄ β£ (Β¬ (π΄ β π¦) β Fin β¨ (π¦ = β β¨ π¦ = π΄))} |
11 | 10 | topdifinffinlem 36879 | 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 845 = wceq 1533 β wcel 2098 {crab 3419 β cdif 3938 β c0 4319 π« cpw 4599 βcfv 6543 Fincfn 8957 TopOnctopon 22825 |
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 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7866 df-1o 8480 df-en 8958 df-fin 8961 df-topgen 17419 df-top 22809 df-topon 22826 |
This theorem is referenced by: topdifinf 36881 |
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