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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 513 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞)) |
4 | xrnepnf 13040 | . . 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 | pm2.53 850 | . 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ℝcr 11051 +∞cpnf 11187 -∞cmnf 11188 ℝ*cxr 11189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-nel 3051 df-rab 3409 df-v 3448 df-un 3916 df-in 3918 df-ss 3928 df-pw 4563 df-sn 4588 df-pr 4590 df-uni 4867 df-pnf 11192 df-mnf 11193 df-xr 11194 |
This theorem is referenced by: xlimliminflimsup 44110 |
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