![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elioc1 | Structured version Visualization version GIF version |
Description: Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
elioc1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioc 13362 | . 2 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | elixx1 13366 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 class class class wbr 5148 (class class class)co 7420 ℝ*cxr 11278 < clt 11279 ≤ cle 11280 (,]cioc 13358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-xr 11283 df-ioc 13362 |
This theorem is referenced by: ubioc1 13410 elioc2 13420 leordtvallem1 23127 pnfnei 23137 mnfnei 23138 xrge0tsms 24763 lhop1 25960 xrlimcnp 26913 iocinioc2 32560 xrge0tsmsd 32784 xrge0iifcnv 33534 lmxrge0 33553 iocinico 42640 rfcnpre4 44396 iocgtlb 44887 iocleub 44888 eliocd 44892 lenelioc 44921 eenglngeehlnmlem1 47810 |
Copyright terms: Public domain | W3C validator |