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Mirrors > Home > MPE Home > Th. List > xrre2 | Structured version Visualization version 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 13065 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
2 | 1 | adantr 482 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ≤ 𝐴) |
3 | mnfxr 11222 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
4 | xrlelttr 13086 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) | |
5 | 3, 4 | mp3an1 1449 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) |
6 | 2, 5 | mpand 694 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
7 | 6 | 3adant3 1133 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
8 | pnfge 13061 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → 𝐶 ≤ +∞) | |
9 | 8 | adantl 483 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ≤ +∞) |
10 | pnfxr 11219 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
11 | xrltletr 13087 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) | |
12 | 10, 11 | mp3an3 1451 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) |
13 | 9, 12 | mpan2d 693 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
14 | 13 | 3adant1 1131 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
15 | 7, 14 | anim12d 610 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
16 | xrrebnd 13098 | . . . 4 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) | |
17 | 16 | 3ad2ant2 1135 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
18 | 15, 17 | sylibrd 259 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ)) |
19 | 18 | imp 408 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5111 ℝcr 11060 +∞cpnf 11196 -∞cmnf 11197 ℝ*cxr 11198 < clt 11199 ≤ cle 11200 |
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 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-pre-lttri 11135 ax-pre-lttrn 11136 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 |
This theorem is referenced by: elioore 13305 xrsdsreclblem 20881 pnfnei 22609 mnfnei 22610 tgioo 24197 ovolunnul 24902 icombl 24966 ioombl 24967 ioorcl2 24974 volivth 25009 dvferm2lem 25388 itg2gt0cn 36207 iccpartipre 45715 |
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