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| Mirrors > Home > MPE Home > Th. List > ordtopn3 | Structured version Visualization version GIF version | ||
| Description: An open interval (π΄, π΅) is open. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| Ref | Expression |
|---|---|
| ordttopon.3 | β’ π = dom π |
| Ref | Expression |
|---|---|
| ordtopn3 | β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ (Β¬ π₯π π΄ β§ Β¬ π΅π π₯)} β (ordTopβπ )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inrab 4305 | . 2 β’ ({π₯ β π β£ Β¬ π₯π π΄} β© {π₯ β π β£ Β¬ π΅π π₯}) = {π₯ β π β£ (Β¬ π₯π π΄ β§ Β¬ π΅π π₯)} | |
| 2 | ordttopon.3 | . . . . . 6 β’ π = dom π | |
| 3 | 2 | ordttopon 22917 | . . . . 5 β’ (π β π β (ordTopβπ ) β (TopOnβπ)) |
| 4 | 3 | 3ad2ant1 1131 | . . . 4 β’ ((π β π β§ π΄ β π β§ π΅ β π) β (ordTopβπ ) β (TopOnβπ)) |
| 5 | topontop 22635 | . . . 4 β’ ((ordTopβπ ) β (TopOnβπ) β (ordTopβπ ) β Top) | |
| 6 | 4, 5 | syl 17 | . . 3 β’ ((π β π β§ π΄ β π β§ π΅ β π) β (ordTopβπ ) β Top) |
| 7 | 2 | ordtopn1 22918 | . . . 4 β’ ((π β π β§ π΄ β π) β {π₯ β π β£ Β¬ π₯π π΄} β (ordTopβπ )) |
| 8 | 7 | 3adant3 1130 | . . 3 β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ Β¬ π₯π π΄} β (ordTopβπ )) |
| 9 | 2 | ordtopn2 22919 | . . . 4 β’ ((π β π β§ π΅ β π) β {π₯ β π β£ Β¬ π΅π π₯} β (ordTopβπ )) |
| 10 | 9 | 3adant2 1129 | . . 3 β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ Β¬ π΅π π₯} β (ordTopβπ )) |
| 11 | inopn 22621 | . . 3 β’ (((ordTopβπ ) β Top β§ {π₯ β π β£ Β¬ π₯π π΄} β (ordTopβπ ) β§ {π₯ β π β£ Β¬ π΅π π₯} β (ordTopβπ )) β ({π₯ β π β£ Β¬ π₯π π΄} β© {π₯ β π β£ Β¬ π΅π π₯}) β (ordTopβπ )) | |
| 12 | 6, 8, 10, 11 | syl3anc 1369 | . 2 β’ ((π β π β§ π΄ β π β§ π΅ β π) β ({π₯ β π β£ Β¬ π₯π π΄} β© {π₯ β π β£ Β¬ π΅π π₯}) β (ordTopβπ )) |
| 13 | 1, 12 | eqeltrrid 2836 | 1 β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ (Β¬ π₯π π΄ β§ Β¬ π΅π π₯)} β (ordTopβπ )) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 {crab 3430 β© cin 3946 class class class wbr 5147 dom cdm 5675 βcfv 6542 ordTopcordt 17449 Topctop 22615 TopOnctopon 22632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7858 df-1o 8468 df-er 8705 df-en 8942 df-fin 8945 df-fi 9408 df-topgen 17393 df-ordt 17451 df-top 22616 df-topon 22633 df-bases 22669 |
| This theorem is referenced by: (None) |
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