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Mirrors > Home > MPE Home > Th. List > ndmioo | Structured version Visualization version GIF version |
Description: The open interval function's value is empty outside of its domain. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ndmioo | ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioo 13222 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | ixxf 13228 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
3 | 2 | fdmi 6677 | . 2 ⊢ dom (,) = (ℝ* × ℝ*) |
4 | 3 | ndmov 7532 | 1 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∅c0 4280 𝒫 cpw 4558 × cxp 5629 (class class class)co 7351 ℝ*cxr 11146 < clt 11147 (,)cioo 13218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-xr 11151 df-ioo 13222 |
This theorem is referenced by: iooid 13246 eliooxr 13276 iccssioo2 13291 ioombl 24881 mbfima 24946 dvferm1lem 25300 dvferm2lem 25302 dvferm 25304 dvivthlem1 25324 |
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