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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 9703 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7937 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2355 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | ltpnf 9707 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | notnot 619 | . . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞) |
7 | 3, 6 | 2falsed 692 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
8 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
9 | pnfxr 7942 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9706 | . . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ +∞ < +∞ |
12 | breq1 3979 | . . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 665 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 8, 13 | 2thd 174 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
15 | mnfnepnf 7945 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
16 | 15 | neii 2336 | . . . . 5 ⊢ ¬ -∞ = +∞ |
17 | eqeq1 2171 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
18 | 16, 17 | mtbiri 665 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
19 | mnfltpnf 9712 | . . . . . . 7 ⊢ -∞ < +∞ | |
20 | breq1 3979 | . . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
21 | 19, 20 | mpbiri 167 | . . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
22 | 21 | necon3bi 2384 | . . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞) |
23 | 22 | necon2bi 2389 | . . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞) |
24 | 18, 23 | 2falsed 692 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
25 | 7, 14, 24 | 3jaoi 1292 | . 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 966 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ℝcr 7743 +∞cpnf 7921 -∞cmnf 7922 ℝ*cxr 7923 < clt 7924 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 |
This theorem is referenced by: npnflt 9742 xgepnf 9743 xrmaxiflemlub 11175 |
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