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Mirrors > Home > ILE Home > Th. List > npnflt | GIF version |
Description: An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.) |
Ref | Expression |
---|---|
npnflt | ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltpnft 9597 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
2 | 1 | biimpd 143 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ → ¬ 𝐴 < +∞)) |
3 | 2 | necon2ad 2365 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ → 𝐴 ≠ +∞)) |
4 | ltpnf 9567 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | 4 | adantl 275 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞) |
6 | simpr 109 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 = +∞) | |
7 | simplr 519 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 ≠ +∞) | |
8 | 6, 7 | pm2.21ddne 2391 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 < +∞) |
9 | mnfltpnf 9571 | . . . . . 6 ⊢ -∞ < +∞ | |
10 | breq1 3932 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
11 | 9, 10 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
12 | 11 | adantl 275 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = -∞) → 𝐴 < +∞) |
13 | elxr 9563 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
14 | 13 | biimpi 119 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
15 | 14 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
16 | 5, 8, 12, 15 | mpjao3dan 1285 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
17 | 16 | ex 114 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≠ +∞ → 𝐴 < +∞)) |
18 | 3, 17 | impbid 128 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 class class class wbr 3929 ℝcr 7619 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 < clt 7800 |
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-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 |
This theorem is referenced by: xlt2add 9663 xrmaxadd 11030 |
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