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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 13104 | . . . 4 ⊢ (0(,)0) = ∅ | |
3 | ioof 13176 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
4 | ffn 6597 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
6 | 0xr 11021 | . . . . 5 ⊢ 0 ∈ ℝ* | |
7 | fnovrn 7439 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 0 ∈ ℝ* ∧ 0 ∈ ℝ*) → (0(,)0) ∈ ran (,)) | |
8 | 5, 6, 6, 7 | mp3an 1460 | . . . 4 ⊢ (0(,)0) ∈ ran (,) |
9 | 2, 8 | eqeltrri 2838 | . . 3 ⊢ ∅ ∈ ran (,) |
10 | 1, 9 | eqeltrdi 2849 | . 2 ⊢ ((𝐴(,)𝐵) = ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
11 | n0 4286 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
12 | eliooxr 13134 | . . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
13 | fnovrn 7439 | . . . . . 6 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) | |
14 | 5, 13 | mp3an1 1447 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
16 | 15 | exlimiv 1937 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
17 | 11, 16 | sylbi 216 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
18 | 10, 17 | pm2.61ine 3030 | 1 ⊢ (𝐴(,)𝐵) ∈ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 ≠ wne 2945 ∅c0 4262 𝒫 cpw 4539 × cxp 5587 ran crn 5590 Fn wfn 6426 ⟶wf 6427 (class class class)co 7269 ℝcr 10869 0cc0 10870 ℝ*cxr 11007 (,)cioo 13076 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-sup 9177 df-inf 9178 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12580 df-q 12686 df-ioo 13080 |
This theorem is referenced by: iooordt 22364 iooretop 23925 blssioo 23954 xrtgioo 23965 ioorinv2 24735 ioorinv 24736 uniioombllem2a 24742 ismbf 24788 |
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