Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9798 | . . . . . . 7 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx3g 9805 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
3 | 2 | biimpi 119 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
4 | 3 | simpld 111 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 996 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 108 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ) | |
8 | 3 | simprd 113 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 8 | simpld 111 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) |
10 | 9 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) |
11 | 4 | simp2d 995 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) |
12 | 11 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 7930 | . . . 4 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → +∞ ∈ ℝ*) |
15 | 8 | simprd 113 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵) |
16 | 15 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) |
17 | pnfge 9696 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
18 | 11, 17 | syl 14 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ≤ +∞) |
19 | 18 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ≤ +∞) |
20 | 6, 12, 14, 16, 19 | xrltletrd 9715 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < +∞) |
21 | xrre3 9726 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < +∞)) → 𝐶 ∈ ℝ) | |
22 | 6, 7, 10, 20, 21 | syl22anc 1221 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 2128 class class class wbr 3965 (class class class)co 5824 ℝcr 7731 +∞cpnf 7909 ℝ*cxr 7911 < clt 7912 ≤ cle 7913 [,)cico 9794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-ico 9798 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |