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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 12174 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | |
2 | 1 | adantr 466 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ≤ 𝐴) |
3 | mnfxr 10302 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
4 | xrlelttr 12192 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) | |
5 | 3, 4 | mp3an1 1559 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵)) |
6 | 2, 5 | mpand 675 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
7 | 6 | 3adant3 1126 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < 𝐵 → -∞ < 𝐵)) |
8 | pnfge 12169 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ* → 𝐶 ≤ +∞) | |
9 | 8 | adantl 467 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐶 ≤ +∞) |
10 | pnfxr 10298 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
11 | xrltletr 12193 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) | |
12 | 10, 11 | mp3an3 1561 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞)) |
13 | 9, 12 | mpan2d 674 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
14 | 13 | 3adant1 1124 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 < 𝐶 → 𝐵 < +∞)) |
15 | 7, 14 | anim12d 596 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
16 | xrrebnd 12204 | . . . 4 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) | |
17 | 16 | 3ad2ant2 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞))) |
18 | 15, 17 | sylibrd 249 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ)) |
19 | 18 | imp 393 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 ∈ wcel 2145 class class class wbr 4787 ℝcr 10141 +∞cpnf 10277 -∞cmnf 10278 ℝ*cxr 10279 < clt 10280 ≤ cle 10281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-pre-lttri 10216 ax-pre-lttrn 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 |
This theorem is referenced by: elioore 12410 xrsdsreclblem 20007 pnfnei 21245 mnfnei 21246 tgioo 22819 ovolunnul 23488 icombl 23552 ioombl 23553 ioorcl2 23560 volivth 23595 dvferm2lem 23969 itg2gt0cn 33796 iccpartipre 41880 |
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