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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 9754 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
2 | ffn 5272 | . . 3 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (,) Fn (ℝ* × ℝ*) |
4 | fnovrn 5918 | . 2 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) | |
5 | 3, 4 | mp3an1 1302 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 𝒫 cpw 3510 × cxp 4537 ran crn 4540 Fn wfn 5118 ⟶wf 5119 (class class class)co 5774 ℝcr 7619 ℝ*cxr 7799 (,)cioo 9671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-ioo 9675 |
This theorem is referenced by: iooretopg 12697 blssioo 12714 tgioo 12715 |
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