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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 9404 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7685 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2288 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | 3 | olcd 694 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) |
5 | df-dc 787 | . . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) | |
6 | 4, 5 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞) |
7 | orc 674 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) | |
8 | 7, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞) |
9 | mnfnepnf 7693 | . . . . . . 7 ⊢ -∞ ≠ +∞ | |
10 | 9 | neii 2269 | . . . . . 6 ⊢ ¬ -∞ = +∞ |
11 | eqeq1 2106 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
12 | 10, 11 | mtbiri 641 | . . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
13 | 12 | olcd 694 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞)) |
14 | 13, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞) |
15 | 6, 8, 14 | 3jaoi 1249 | . 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 670 DECID wdc 786 ∨ w3o 929 = wceq 1299 ∈ wcel 1448 ℝcr 7499 +∞cpnf 7669 -∞cmnf 7670 ℝ*cxr 7671 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-un 4293 ax-cnex 7586 ax-resscn 7587 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-rex 2381 df-rab 2384 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-pnf 7674 df-mnf 7675 df-xr 7676 |
This theorem is referenced by: xaddf 9468 xaddval 9469 xaddpnf1 9470 xaddcom 9485 xnegdi 9492 xleadd1a 9497 xlesubadd 9507 xrmaxiflemcl 10853 xrmaxifle 10854 xrmaxiflemab 10855 xrmaxiflemlub 10856 xrmaxiflemcom 10857 xrmaxadd 10869 xblss2ps 12332 xblss2 12333 |
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