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Mirrors > Home > ILE Home > Th. List > ioorebasg | GIF version |
Description: Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.) |
Ref | Expression |
---|---|
ioorebasg | ⊢ ((A ∈ ℝ* ∧ B ∈ ℝ*) → (A(,)B) ∈ ran (,)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioof 8610 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 4989 | . . 3 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (,) Fn (ℝ* × ℝ*) |
4 | fnovrn 5590 | . 2 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ A ∈ ℝ* ∧ B ∈ ℝ*) → (A(,)B) ∈ ran (,)) | |
5 | 3, 4 | mp3an1 1218 | 1 ⊢ ((A ∈ ℝ* ∧ B ∈ ℝ*) → (A(,)B) ∈ ran (,)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 𝒫 cpw 3351 × cxp 4286 ran crn 4289 Fn wfn 4840 ⟶wf 4841 (class class class)co 5455 ℝcr 6710 ℝ*cxr 6856 (,)cioo 8527 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-po 4024 df-iso 4025 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-ioo 8531 |
This theorem is referenced by: (None) |
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