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Mirrors > Home > ILE Home > Th. List > xrpnfdc | GIF version |
Description: An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
Ref | Expression |
---|---|
xrpnfdc | ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9566 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7816 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2329 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | 3 | olcd 723 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) |
5 | df-dc 820 | . . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) | |
6 | 4, 5 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞) |
7 | orc 701 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) | |
8 | 7, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞) |
9 | mnfnepnf 7824 | . . . . . . 7 ⊢ -∞ ≠ +∞ | |
10 | 9 | neii 2310 | . . . . . 6 ⊢ ¬ -∞ = +∞ |
11 | eqeq1 2146 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
12 | 10, 11 | mtbiri 664 | . . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
13 | 12 | olcd 723 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) |
14 | 13, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞) |
15 | 6, 8, 14 | 3jaoi 1281 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = +∞) |
16 | 1, 15 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 DECID wdc 819 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ℝcr 7622 +∞cpnf 7800 -∞cmnf 7801 ℝ*cxr 7802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-un 4355 ax-cnex 7714 ax-resscn 7715 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-pnf 7805 df-mnf 7806 df-xr 7807 |
This theorem is referenced by: xaddf 9630 xaddval 9631 xaddpnf1 9632 xaddcom 9647 xnegdi 9654 xleadd1a 9659 xlesubadd 9669 xrmaxiflemcl 11017 xrmaxifle 11018 xrmaxiflemab 11019 xrmaxiflemlub 11020 xrmaxiflemcom 11021 xrmaxadd 11033 xblss2ps 12576 xblss2 12577 |
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