![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ioorebas | Structured version Visualization version GIF version |
Description: Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
ioorebas | ⊢ (𝐴(,)𝐵) ∈ ran (,) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝐴(,)𝐵) = ∅ → (𝐴(,)𝐵) = ∅) | |
2 | iooid 13401 | . . . 4 ⊢ (0(,)0) = ∅ | |
3 | ioof 13473 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
4 | ffn 6727 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
6 | 0xr 11307 | . . . . 5 ⊢ 0 ∈ ℝ* | |
7 | fnovrn 7600 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 0 ∈ ℝ* ∧ 0 ∈ ℝ*) → (0(,)0) ∈ ran (,)) | |
8 | 5, 6, 6, 7 | mp3an 1457 | . . . 4 ⊢ (0(,)0) ∈ ran (,) |
9 | 2, 8 | eqeltrri 2822 | . . 3 ⊢ ∅ ∈ ran (,) |
10 | 1, 9 | eqeltrdi 2833 | . 2 ⊢ ((𝐴(,)𝐵) = ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
11 | n0 4348 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
12 | eliooxr 13431 | . . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
13 | fnovrn 7600 | . . . . . 6 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) | |
14 | 5, 13 | mp3an1 1444 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
16 | 15 | exlimiv 1925 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
17 | 11, 16 | sylbi 216 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
18 | 10, 17 | pm2.61ine 3014 | 1 ⊢ (𝐴(,)𝐵) ∈ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 ∅c0 4324 𝒫 cpw 4606 × cxp 5679 ran crn 5682 Fn wfn 6548 ⟶wf 6549 (class class class)co 7423 ℝcr 11153 0cc0 11154 ℝ*cxr 11293 (,)cioo 13373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-n0 12520 df-z 12606 df-uz 12870 df-q 12980 df-ioo 13377 |
This theorem is referenced by: iooordt 23204 iooretop 24765 blssioo 24794 xrtgioo 24805 ioorinv2 25587 ioorinv 25588 uniioombllem2a 25594 ismbf 25640 |
Copyright terms: Public domain | W3C validator |