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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 4286 | . 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 22596 | . . . . 5 β’ (π β π β (ordTopβπ ) β (TopOnβπ)) |
| 4 | 3 | 3ad2ant1 1133 | . . . 4 β’ ((π β π β§ π΄ β π β§ π΅ β π) β (ordTopβπ ) β (TopOnβπ)) |
| 5 | topontop 22314 | . . . 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 22597 | . . . 4 β’ ((π β π β§ π΄ β π) β {π₯ β π β£ Β¬ π₯π π΄} β (ordTopβπ )) |
| 8 | 7 | 3adant3 1132 | . . 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 22598 | . . . 4 β’ ((π β π β§ π΅ β π) β {π₯ β π β£ Β¬ π΅π π₯} β (ordTopβπ )) |
| 10 | 9 | 3adant2 1131 | . . 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 22300 | . . 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 1371 | . 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 2837 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 {crab 3418 β© cin 3927 class class class wbr 5125 dom cdm 5653 βcfv 6516 ordTopcordt 17410 Topctop 22294 TopOnctopon 22311 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-om 7823 df-1o 8432 df-er 8670 df-en 8906 df-fin 8909 df-fi 9371 df-topgen 17354 df-ordt 17412 df-top 22295 df-topon 22312 df-bases 22348 |
| This theorem is referenced by: (None) |
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