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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 511 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xrnemnf 13156 | . . 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 2942 | . 2 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
8 | pm2.53 851 | . . 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 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ℝcr 11151 +∞cpnf 11289 -∞cmnf 11290 ℝ*cxr 11291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 |
This theorem is referenced by: infxr 45316 infleinflem2 45320 xrralrecnnge 45339 xrre4 45360 supminfxr2 45418 xrpnf 45435 climxrrelem 45704 climxrre 45705 liminflimsupxrre 45772 ioorrnopnxrlem 46261 pimiooltgt 46665 smfpimltxr 46702 smfpimgtxr 46735 |
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