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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 13161 | . . 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 1539 ∈ wcel 2107 ≠ wne 2939 ℝcr 11155 +∞cpnf 11293 -∞cmnf 11294 ℝ*cxr 11295 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-nel 3046 df-rab 3436 df-v 3481 df-un 3955 df-in 3957 df-ss 3967 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 df-pnf 11298 df-mnf 11299 df-xr 11300 | 
| This theorem is referenced by: xlimliminflimsup 45882 | 
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