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Mirrors > Home > MPE Home > Th. List > Mathboxes > reopn | Structured version Visualization version GIF version |
Description: The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
reopn | ⊢ ℝ ∈ (topGen‘ran (,)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 23526 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | uniretop 23527 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
3 | 2 | topopn 21669 | . 2 ⊢ ((topGen‘ran (,)) ∈ Top → ℝ ∈ (topGen‘ran (,))) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ℝ ∈ (topGen‘ran (,)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ran crn 5536 ‘cfv 6349 ℝcr 10626 (,)cioo 12833 topGenctg 16826 Topctop 21656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-pre-lttri 10701 ax-pre-lttrn 10702 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-ioo 12837 df-topgen 16832 df-top 21657 df-bases 21709 |
This theorem is referenced by: fperdvper 43042 dirkeritg 43225 etransclem2 43359 etransclem23 43380 etransclem35 43392 etransclem38 43395 etransclem39 43396 etransclem44 43401 etransclem45 43402 etransclem47 43404 |
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