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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoreelrn | Structured version Visualization version GIF version |
Description: A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.) |
Ref | Expression |
---|---|
icoreelrn.1 | ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) |
Ref | Expression |
---|---|
icoreelrn | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoreval 36172 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | |
2 | simpl 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
3 | simpr 486 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
4 | df-ico 13326 | . . . . . 6 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
5 | 4 | ixxf 13330 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
6 | ffun 6717 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
7 | 5, 6 | mp1i 13 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → Fun [,)) |
8 | rexpssxrxp 11255 | . . . . . 6 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
9 | 5 | fdmi 6726 | . . . . . 6 ⊢ dom [,) = (ℝ* × ℝ*) |
10 | 8, 9 | sseqtrri 4018 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ dom [,) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ × ℝ) ⊆ dom [,)) |
12 | 2, 3, 7, 11 | elovimad 7452 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ ([,) “ (ℝ × ℝ))) |
13 | icoreelrn.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
14 | 12, 13 | eleqtrrdi 2845 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ 𝐼) |
15 | 1, 14 | eqeltrrd 2835 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3947 𝒫 cpw 4601 class class class wbr 5147 × cxp 5673 dom cdm 5675 “ cima 5678 Fun wfun 6534 ⟶wf 6536 (class class class)co 7404 ℝcr 11105 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ico 13326 |
This theorem is referenced by: relowlssretop 36182 |
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