Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrre4 | Structured version Visualization version GIF version |
Description: An extended real is real iff it is not an infinty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
xrre4 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renemnf 10678 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞) |
3 | renepnf 10677 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
4 | 3 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞) |
5 | 2, 4 | jca 512 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) |
6 | 5 | ex 413 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) |
7 | simpl 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ*) | |
8 | simprl 767 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞) | |
9 | simprr 769 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞) | |
10 | 7, 8, 9 | xrred 41509 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ) |
11 | 10 | ex 413 | . 2 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)) |
12 | 6, 11 | impbid 213 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 |
This theorem is referenced by: limsupre2lem 41881 |
Copyright terms: Public domain | W3C validator |