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 514 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xrnemnf 12515 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
5 | 3, 4 | sylib 220 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | xrred.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
7 | 6 | neneqd 3023 | . 2 ⊢ (𝜑 → ¬ 𝐴 = +∞) |
8 | pm2.53 847 | . . 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 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ℝcr 10538 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 |
This theorem is referenced by: infxr 41642 infleinflem2 41646 xrralrecnnge 41669 xrre4 41692 supminfxr2 41752 xrpnf 41769 climxrrelem 42037 climxrre 42038 liminflimsupxrre 42105 ioorrnopnxrlem 42598 pimiooltgt 42996 smfpimltxr 43031 smfpimgtxr 43063 |
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