Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnmnfpnf | Structured version Visualization version GIF version |
Description: An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
xrnmnfpnf.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrnmnfpnf.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) |
xrnmnfpnf.3 | ⊢ (𝜑 → 𝐴 ≠ -∞) |
Ref | Expression |
---|---|
xrnmnfpnf | ⊢ (𝜑 → 𝐴 = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnmnfpnf.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrnmnfpnf.3 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ -∞) | |
3 | 1, 2 | jca 515 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xrnemnf 12588 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
5 | 3, 4 | sylib 221 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | xrnmnfpnf.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 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ℝcr 10607 +∞cpnf 10743 -∞cmnf 10744 ℝ*cxr 10745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 |
This theorem is referenced by: infxr 42428 dfxlim2v 42914 xlimliminflimsup 42929 |
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