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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 13111 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ≤ 𝐴) |
3 | mnfxr 11268 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
4 | xrlelttr 13132 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) | |
5 | 3, 4 | mp3an1 1444 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) |
6 | 2, 5 | mpand 692 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
7 | 6 | 3adant3 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
8 | pnfge 13107 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → 𝐶 ≤ +∞) | |
9 | 8 | adantl 481 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ≤ +∞) |
10 | pnfxr 11265 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
11 | xrltletr 13133 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) | |
12 | 10, 11 | mp3an3 1446 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) |
13 | 9, 12 | mpan2d 691 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
14 | 13 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
15 | 7, 14 | anim12d 608 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
16 | xrrebnd 13144 | . . . 4 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) | |
17 | 16 | 3ad2ant2 1131 | . . 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 406 | 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 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5138 ℝcr 11105 +∞cpnf 11242 -∞cmnf 11243 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 |
This theorem is referenced by: elioore 13351 xrsdsreclblem 21275 pnfnei 23046 mnfnei 23047 tgioo 24634 ovolunnul 25351 icombl 25415 ioombl 25416 ioorcl2 25423 volivth 25458 dvferm2lem 25840 itg2gt0cn 37033 iccpartipre 46574 |
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