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Mirrors > Home > ILE Home > Th. List > elicore | GIF version |
Description: A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elicore | ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 9963 | . . . . . . 7 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx3g 9970 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
3 | 2 | biimpi 120 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
4 | 3 | simpld 112 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1013 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 277 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 109 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ) | |
8 | 3 | simprd 114 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 8 | simpld 112 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) |
10 | 9 | adantl 277 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) |
11 | 4 | simp2d 1012 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) |
12 | 11 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 8074 | . . . 4 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → +∞ ∈ ℝ*) |
15 | 8 | simprd 114 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵) |
16 | 15 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) |
17 | pnfge 9858 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
18 | 11, 17 | syl 14 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ≤ +∞) |
19 | 18 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ≤ +∞) |
20 | 6, 12, 14, 16, 19 | xrltletrd 9880 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < +∞) |
21 | xrre3 9891 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < +∞)) → 𝐶 ∈ ℝ) | |
22 | 6, 7, 10, 20, 21 | syl22anc 1250 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 ∈ wcel 2164 class class class wbr 4030 (class class class)co 5919 ℝcr 7873 +∞cpnf 8053 ℝ*cxr 8055 < clt 8056 ≤ cle 8057 [,)cico 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-ico 9963 |
This theorem is referenced by: (None) |
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