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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 12756 | . . . 4 ⊢ (0(,)0) = ∅ | |
3 | ioof 12825 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
4 | ffn 6508 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
6 | 0xr 10677 | . . . . 5 ⊢ 0 ∈ ℝ* | |
7 | fnovrn 7312 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 0 ∈ ℝ* ∧ 0 ∈ ℝ*) → (0(,)0) ∈ ran (,)) | |
8 | 5, 6, 6, 7 | mp3an 1452 | . . . 4 ⊢ (0(,)0) ∈ ran (,) |
9 | 2, 8 | eqeltrri 2910 | . . 3 ⊢ ∅ ∈ ran (,) |
10 | 1, 9 | syl6eqel 2921 | . 2 ⊢ ((𝐴(,)𝐵) = ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
11 | n0 4309 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
12 | eliooxr 12785 | . . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
13 | fnovrn 7312 | . . . . . 6 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) | |
14 | 5, 13 | mp3an1 1439 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
16 | 15 | exlimiv 1922 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
17 | 11, 16 | sylbi 218 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
18 | 10, 17 | pm2.61ine 3100 | 1 ⊢ (𝐴(,)𝐵) ∈ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3016 ∅c0 4290 𝒫 cpw 4537 × cxp 5547 ran crn 5550 Fn wfn 6344 ⟶wf 6345 (class class class)co 7145 ℝcr 10525 0cc0 10526 ℝ*cxr 10663 (,)cioo 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-ioo 12732 |
This theorem is referenced by: iooordt 21755 iooretop 23303 blssioo 23332 xrtgioo 23343 ioorinv2 24105 ioorinv 24106 uniioombllem2a 24112 ismbf 24158 |
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