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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnpnfmnf | Structured version Visualization version GIF version |
Description: An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xrnpnfmnf.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrnpnfmnf.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) |
xrnpnfmnf.3 | ⊢ (𝜑 → 𝐴 ≠ +∞) |
Ref | Expression |
---|---|
xrnpnfmnf | ⊢ (𝜑 → 𝐴 = -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnpnfmnf.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrnpnfmnf.3 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ +∞) | |
3 | 1, 2 | jca 511 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞)) |
4 | xrnepnf 13158 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) | |
5 | 3, 4 | sylib 218 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
6 | xrnpnfmnf.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) | |
7 | pm2.53 851 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞)) | |
8 | 5, 6, 7 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 = -∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ℝcr 11152 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-nel 3045 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-pnf 11295 df-mnf 11296 df-xr 11297 |
This theorem is referenced by: xlimliminflimsup 45818 |
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