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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 9846 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
2 | 1 | biimpd 144 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ → ¬ 𝐴 < +∞)) |
3 | 2 | necon2ad 2417 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ → 𝐴 ≠ +∞)) |
4 | ltpnf 9812 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | 4 | adantl 277 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞) |
6 | simpr 110 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 = +∞) | |
7 | simplr 528 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 ≠ +∞) | |
8 | 6, 7 | pm2.21ddne 2443 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = +∞) → 𝐴 < +∞) |
9 | mnfltpnf 9817 | . . . . . 6 ⊢ -∞ < +∞ | |
10 | breq1 4021 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
11 | 9, 10 | mpbiri 168 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
12 | 11 | adantl 277 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ 𝐴 = -∞) → 𝐴 < +∞) |
13 | elxr 9808 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
14 | 13 | biimpi 120 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
15 | 14 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
16 | 5, 8, 12, 15 | mpjao3dan 1318 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → 𝐴 < +∞) |
17 | 16 | ex 115 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≠ +∞ → 𝐴 < +∞)) |
18 | 3, 17 | impbid 129 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 class class class wbr 4018 ℝcr 7841 +∞cpnf 8020 -∞cmnf 8021 ℝ*cxr 8022 < clt 8023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-pre-ltirr 7954 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 |
This theorem is referenced by: xlt2add 9912 xrmaxadd 11304 |
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