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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 12179 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | ixxf 12185 | . . 3 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
3 | 2 | fdmi 6052 | . 2 ⊢ dom (,) = (ℝ* × ℝ*) |
4 | 3 | ndmov 6818 | 1 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 𝒫 cpw 4158 × cxp 5112 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-xr 10078 df-ioo 12179 |
This theorem is referenced by: iooid 12203 eliooxr 12232 iccssioo2 12246 ioombl 23333 mbfima 23399 dvferm1lem 23747 dvferm2lem 23749 dvferm 23751 dvivthlem1 23771 |
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