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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliocre | Structured version Visualization version GIF version |
Description: A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eliocre | ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioc 13332 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | elixx3g 13340 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴(,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
4 | 3 | simpld 494 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1141 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 482 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐵 ∈ ℝ) | |
8 | mnfxr 11272 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ ∈ ℝ*) |
10 | 4 | simp1d 1139 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 ∈ ℝ*) |
11 | mnfle 13117 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ ≤ 𝐴) |
13 | 3 | simprd 495 | . . . . 5 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → (𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
14 | 13 | simpld 494 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐶) |
15 | 9, 10, 5, 12, 14 | xrlelttrd 13142 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ < 𝐶) |
16 | 15 | adantl 481 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → -∞ < 𝐶) |
17 | 13 | simprd 495 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ≤ 𝐵) |
18 | 17 | adantl 481 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) |
19 | xrre 13151 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ) | |
20 | 6, 7, 16, 18, 19 | syl22anc 836 | 1 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 -∞cmnf 11247 ℝ*cxr 11248 < clt 11249 ≤ cle 11250 (,]cioc 13328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ioc 13332 |
This theorem is referenced by: (None) |
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