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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 13270 | . . . . . . 7 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | elixx3g 13278 | . . . . . 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 496 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1145 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 483 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 484 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐵 ∈ ℝ) | |
8 | mnfxr 11213 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ ∈ ℝ*) |
10 | 4 | simp1d 1143 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 ∈ ℝ*) |
11 | mnfle 13056 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ ≤ 𝐴) |
13 | 3 | simprd 497 | . . . . 5 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → (𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
14 | 13 | simpld 496 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐴 < 𝐶) |
15 | 9, 10, 5, 12, 14 | xrlelttrd 13080 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → -∞ < 𝐶) |
16 | 15 | adantl 483 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → -∞ < 𝐶) |
17 | 13 | simprd 497 | . . 3 ⊢ (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ≤ 𝐵) |
18 | 17 | adantl 483 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) |
19 | xrre 13089 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ) | |
20 | 6, 7, 16, 18, 19 | syl22anc 838 | 1 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11051 -∞cmnf 11188 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 (,]cioc 13266 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ioc 13270 |
This theorem is referenced by: (None) |
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