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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 12051 | . 2 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | elixx1 12055 | 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 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,]cioc 12047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-xr 9957 df-ioc 12051 |
This theorem is referenced by: ubioc1 12098 elioc2 12107 leordtvallem1 20824 pnfnei 20834 mnfnei 20835 xrge0tsms 22445 lhop1 23581 xrlimcnp 24495 iocinioc2 28931 xrge0tsmsd 29116 xrge0iifcnv 29307 lmxrge0 29326 iocinico 36816 rfcnpre4 38216 iocgtlb 38571 iocleub 38572 eliocd 38577 lenelioc 38610 |
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