| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xrred | Structured version Visualization version GIF version | ||
| Description: An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
| Ref | Expression |
|---|---|
| xrred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| xrred.2 | ⊢ (𝜑 → 𝐴 ≠ -∞) |
| xrred.3 | ⊢ (𝜑 → 𝐴 ≠ +∞) |
| Ref | Expression |
|---|---|
| xrred | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrred.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 2 | xrred.2 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ -∞) | |
| 3 | 1, 2 | jca 511 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
| 4 | xrnemnf 13159 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
| 5 | 3, 4 | sylib 218 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
| 6 | xrred.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
| 7 | 6 | neneqd 2945 | . 2 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
| 8 | pm2.53 852 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞)) | |
| 9 | 8 | con1d 145 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 = +∞ → 𝐴 ∈ ℝ)) |
| 10 | 5, 7, 9 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 |
| This theorem is referenced by: infxr 45378 infleinflem2 45382 xrralrecnnge 45401 xrre4 45422 supminfxr2 45480 xrpnf 45496 climxrrelem 45764 climxrre 45765 liminflimsupxrre 45832 ioorrnopnxrlem 46321 pimiooltgt 46725 smfpimltxr 46762 smfpimgtxr 46795 |
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