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Mirrors > Home > MPE Home > Th. List > ordtcld3 | Structured version Visualization version GIF version |
Description: A closed interval [π΄, π΅] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ordttopon.3 | β’ π = dom π |
Ref | Expression |
---|---|
ordtcld3 | β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ (π΄π π₯ β§ π₯π π΅)} β (Clsdβ(ordTopβπ ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inrab 4286 | . 2 β’ ({π₯ β π β£ π΄π π₯} β© {π₯ β π β£ π₯π π΅}) = {π₯ β π β£ (π΄π π₯ β§ π₯π π΅)} | |
2 | ordttopon.3 | . . . . 5 β’ π = dom π | |
3 | 2 | ordtcld2 22601 | . . . 4 β’ ((π β π β§ π΄ β π) β {π₯ β π β£ π΄π π₯} β (Clsdβ(ordTopβπ ))) |
4 | 3 | 3adant3 1132 | . . 3 β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ π΄π π₯} β (Clsdβ(ordTopβπ ))) |
5 | 2 | ordtcld1 22600 | . . 3 β’ ((π β π β§ π΅ β π) β {π₯ β π β£ π₯π π΅} β (Clsdβ(ordTopβπ ))) |
6 | incld 22446 | . . 3 β’ (({π₯ β π β£ π΄π π₯} β (Clsdβ(ordTopβπ )) β§ {π₯ β π β£ π₯π π΅} β (Clsdβ(ordTopβπ ))) β ({π₯ β π β£ π΄π π₯} β© {π₯ β π β£ π₯π π΅}) β (Clsdβ(ordTopβπ ))) | |
7 | 4, 5, 6 | 3imp3i2an 1345 | . 2 β’ ((π β π β§ π΄ β π β§ π΅ β π) β ({π₯ β π β£ π΄π π₯} β© {π₯ β π β£ π₯π π΅}) β (Clsdβ(ordTopβπ ))) |
8 | 1, 7 | eqeltrrid 2837 | 1 β’ ((π β π β§ π΄ β π β§ π΅ β π) β {π₯ β π β£ (π΄π π₯ β§ π₯π π΅)} β (Clsdβ(ordTopβπ ))) |
Colors of variables: wff setvar class |
Syntax hints: 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 Clsdccld 22419 |
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-iun 4976 df-iin 4977 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 df-cld 22422 |
This theorem is referenced by: iccordt 22617 ordtt1 22782 |
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