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Mirrors > Home > ILE Home > Th. List > nltpnft | GIF version |
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
nltpnft | ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7813 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2329 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | ltpnf 9567 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | notnot 618 | . . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞) |
7 | 3, 6 | 2falsed 691 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
8 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
9 | pnfxr 7818 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9566 | . . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ +∞ < +∞ |
12 | breq1 3932 | . . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 664 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 8, 13 | 2thd 174 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
15 | mnfnepnf 7821 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
16 | 15 | neii 2310 | . . . . 5 ⊢ ¬ -∞ = +∞ |
17 | eqeq1 2146 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
18 | 16, 17 | mtbiri 664 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
19 | mnfltpnf 9571 | . . . . . . 7 ⊢ -∞ < +∞ | |
20 | breq1 3932 | . . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
21 | 19, 20 | mpbiri 167 | . . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
22 | 21 | necon3bi 2358 | . . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞) |
23 | 22 | necon2bi 2363 | . . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞) |
24 | 18, 23 | 2falsed 691 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
25 | 7, 14, 24 | 3jaoi 1281 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
26 | 1, 25 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 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: npnflt 9598 xgepnf 9599 xrmaxiflemlub 11017 |
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