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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 512 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞)) |
4 | xrnepnf 13100 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) | |
5 | 3, 4 | sylib 217 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
6 | xrnpnfmnf.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) | |
7 | pm2.53 849 | . 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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ℝcr 11111 +∞cpnf 11247 -∞cmnf 11248 ℝ*cxr 11249 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-nel 3047 df-rab 3433 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-pw 4604 df-sn 4629 df-pr 4631 df-uni 4909 df-pnf 11252 df-mnf 11253 df-xr 11254 |
This theorem is referenced by: xlimliminflimsup 44657 |
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