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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 34770 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | |
2 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
3 | simpr 488 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
4 | df-ico 12732 | . . . . . 6 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
5 | 4 | ixxf 12736 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
6 | ffun 6490 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
7 | 5, 6 | mp1i 13 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → Fun [,)) |
8 | rexpssxrxp 10675 | . . . . . 6 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
9 | 5 | fdmi 6498 | . . . . . 6 ⊢ dom [,) = (ℝ* × ℝ*) |
10 | 8, 9 | sseqtrri 3952 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ dom [,) |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ × ℝ) ⊆ dom [,)) |
12 | 2, 3, 7, 11 | elovimad 7183 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ ([,) “ (ℝ × ℝ))) |
13 | icoreelrn.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
14 | 12, 13 | eleqtrrdi 2901 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ 𝐼) |
15 | 1, 14 | eqeltrrd 2891 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 × cxp 5517 dom cdm 5519 “ cima 5522 Fun wfun 6318 ⟶wf 6320 (class class class)co 7135 ℝcr 10525 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 |
This theorem is referenced by: relowlssretop 34780 |
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