![]() |
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 11262 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞) |
3 | renepnf 11261 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
4 | 3 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞) |
5 | 2, 4 | jca 511 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) |
6 | 5 | ex 412 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) |
7 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ*) | |
8 | simprl 768 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞) | |
9 | simprr 770 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞) | |
10 | 7, 8, 9 | xrred 44620 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ) |
11 | 10 | ex 412 | . 2 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ)) |
12 | 6, 11 | impbid 211 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ≠ wne 2932 ℝcr 11106 +∞cpnf 11244 -∞cmnf 11245 ℝ*cxr 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 |
This theorem is referenced by: limsupre2lem 44985 |
Copyright terms: Public domain | W3C validator |