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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 37355 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | |
| 2 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
| 3 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
| 4 | df-ico 13394 | . . . . . 6 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
| 5 | 4 | ixxf 13398 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | 
| 6 | ffun 6738 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
| 7 | 5, 6 | mp1i 13 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → Fun [,)) | 
| 8 | rexpssxrxp 11307 | . . . . . 6 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
| 9 | 5 | fdmi 6746 | . . . . . 6 ⊢ dom [,) = (ℝ* × ℝ*) | 
| 10 | 8, 9 | sseqtrri 4032 | . . . . 5 ⊢ (ℝ × ℝ) ⊆ dom [,) | 
| 11 | 10 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ × ℝ) ⊆ dom [,)) | 
| 12 | 2, 3, 7, 11 | elovimad 7482 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ ([,) “ (ℝ × ℝ))) | 
| 13 | icoreelrn.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
| 14 | 12, 13 | eleqtrrdi 2851 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) ∈ 𝐼) | 
| 15 | 1, 14 | eqeltrrd 2841 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 𝒫 cpw 4599 class class class wbr 5142 × cxp 5682 dom cdm 5684 “ cima 5687 Fun wfun 6554 ⟶wf 6556 (class class class)co 7432 ℝcr 11155 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 [,)cico 13390 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ico 13394 | 
| This theorem is referenced by: relowlssretop 37365 | 
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