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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 9245 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7533 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2276 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | ltpnf 9249 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | notnot 594 | . . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞) |
7 | 3, 6 | 2falsed 653 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
8 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
9 | pnfxr 7538 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9248 | . . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 7 | . . . . 5 ⊢ ¬ +∞ < +∞ |
12 | breq1 3848 | . . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 635 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 8, 13 | 2thd 173 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
15 | mnfnepnf 7541 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
16 | 15 | neii 2257 | . . . . 5 ⊢ ¬ -∞ = +∞ |
17 | eqeq1 2094 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
18 | 16, 17 | mtbiri 635 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
19 | mnfltpnf 9253 | . . . . . . 7 ⊢ -∞ < +∞ | |
20 | breq1 3848 | . . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
21 | 19, 20 | mpbiri 166 | . . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
22 | 21 | necon3bi 2305 | . . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞) |
23 | 22 | necon2bi 2310 | . . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞) |
24 | 18, 23 | 2falsed 653 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
25 | 7, 14, 24 | 3jaoi 1239 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
26 | 1, 25 | sylbi 119 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∨ w3o 923 = wceq 1289 ∈ wcel 1438 class class class wbr 3845 ℝcr 7347 +∞cpnf 7517 -∞cmnf 7518 ℝ*cxr 7519 < clt 7520 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-pre-ltirr 7455 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 |
This theorem is referenced by: (None) |
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