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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 12491 | . . . 4 ⊢ (0(,)0) = ∅ | |
3 | ioof 12560 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
4 | ffn 6278 | . . . . . 6 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (,) Fn (ℝ* × ℝ*) |
6 | 0xr 10403 | . . . . 5 ⊢ 0 ∈ ℝ* | |
7 | fnovrn 7069 | . . . . 5 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 0 ∈ ℝ* ∧ 0 ∈ ℝ*) → (0(,)0) ∈ ran (,)) | |
8 | 5, 6, 6, 7 | mp3an 1591 | . . . 4 ⊢ (0(,)0) ∈ ran (,) |
9 | 2, 8 | eqeltrri 2903 | . . 3 ⊢ ∅ ∈ ran (,) |
10 | 1, 9 | syl6eqel 2914 | . 2 ⊢ ((𝐴(,)𝐵) = ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
11 | n0 4160 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴(,)𝐵)) | |
12 | eliooxr 12520 | . . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) | |
13 | fnovrn 7069 | . . . . . 6 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) | |
14 | 5, 13 | mp3an1 1578 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
16 | 15 | exlimiv 2031 | . . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ∈ ran (,)) |
17 | 11, 16 | sylbi 209 | . 2 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴(,)𝐵) ∈ ran (,)) |
18 | 10, 17 | pm2.61ine 3082 | 1 ⊢ (𝐴(,)𝐵) ∈ ran (,) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ≠ wne 2999 ∅c0 4144 𝒫 cpw 4378 × cxp 5340 ran crn 5343 Fn wfn 6118 ⟶wf 6119 (class class class)co 6905 ℝcr 10251 0cc0 10252 ℝ*cxr 10390 (,)cioo 12463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-ioo 12467 |
This theorem is referenced by: iooordt 21392 iooretop 22939 blssioo 22968 xrtgioo 22979 ioorinv2 23741 ioorinv 23742 uniioombllem2a 23748 ismbf 23794 |
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