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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 12825 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | ixxf 12831 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
3 | 2 | fdmi 6516 | . 2 ⊢ dom (,) = (ℝ* × ℝ*) |
4 | 3 | ndmov 7348 | 1 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∅c0 4211 𝒫 cpw 4488 × cxp 5523 (class class class)co 7170 ℝ*cxr 10752 < clt 10753 (,)cioo 12821 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-xr 10757 df-ioo 12825 |
This theorem is referenced by: iooid 12849 eliooxr 12879 iccssioo2 12894 ioombl 24317 mbfima 24382 dvferm1lem 24736 dvferm2lem 24738 dvferm 24740 dvivthlem1 24760 |
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