Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xrre2 | GIF version |
Description: An extended real between two others is real. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
xrre2 | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfle 8867 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
2 | 1 | adantr 270 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ≤ 𝐴) |
3 | mnfxr 8848 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
4 | xrlelttr 8876 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) | |
5 | 3, 4 | mp3an1 1255 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) |
6 | 2, 5 | mpand 419 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
7 | 6 | 3adant3 958 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
8 | pnfge 8864 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → 𝐶 ≤ +∞) | |
9 | 8 | adantl 271 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ≤ +∞) |
10 | pnfxr 8846 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
11 | xrltletr 8877 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) | |
12 | 10, 11 | mp3an3 1257 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) |
13 | 9, 12 | mpan2d 418 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
14 | 13 | 3adant1 956 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
15 | 7, 14 | anim12d 328 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
16 | xrrebnd 8886 | . . . 4 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) | |
17 | 16 | 3ad2ant2 960 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
18 | 15, 17 | sylibrd 167 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ)) |
19 | 18 | imp 122 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 +∞cpnf 7150 -∞cmnf 7151 ℝ*cxr 7152 < clt 7153 ≤ cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-po 4051 df-iso 4052 df-xp 4369 df-cnv 4371 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 |
This theorem is referenced by: elioore 8935 |
Copyright terms: Public domain | W3C validator |