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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 515 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xrnemnf 12500 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
5 | 3, 4 | sylib 221 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | xrred.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
7 | 6 | neneqd 2992 | . 2 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
8 | pm2.53 848 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞)) | |
9 | 8 | con1d 147 | . 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 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 |
This theorem is referenced by: infxr 41999 infleinflem2 42003 xrralrecnnge 42026 xrre4 42048 supminfxr2 42108 xrpnf 42125 climxrrelem 42391 climxrre 42392 liminflimsupxrre 42459 ioorrnopnxrlem 42948 pimiooltgt 43346 smfpimltxr 43381 smfpimgtxr 43413 |
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