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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 508 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xrnemnf 12198 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
5 | 3, 4 | sylib 210 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | xrred.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
7 | 6 | neneqd 2976 | . 2 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
8 | pm2.53 878 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞)) | |
9 | 8 | con1d 142 | . 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 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ℝcr 10223 +∞cpnf 10360 -∞cmnf 10361 ℝ*cxr 10362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 |
This theorem is referenced by: infxr 40327 infleinflem2 40331 xrralrecnnge 40356 xrre4 40381 supminfxr2 40442 xrpnf 40459 climxrrelem 40725 climxrre 40726 ioorrnopnxrlem 41269 pimiooltgt 41667 smfpimltxr 41702 smfpimgtxr 41734 |
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