![]() |
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 11287 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞) |
3 | renepnf 11286 | . . . . 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 770 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞) | |
9 | simprr 772 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞) | |
10 | 7, 8, 9 | xrred 44719 | . . 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 2099 ≠ wne 2935 ℝcr 11131 +∞cpnf 11269 -∞cmnf 11270 ℝ*cxr 11271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 |
This theorem is referenced by: limsupre2lem 45084 |
Copyright terms: Public domain | W3C validator |