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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 9456 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 7737 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2303 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | ltpnf 9460 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | notnot 601 | . . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞) |
7 | 3, 6 | 2falsed 674 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
8 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
9 | pnfxr 7742 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9459 | . . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 7 | . . . . 5 ⊢ ¬ +∞ < +∞ |
12 | breq1 3898 | . . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 647 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 8, 13 | 2thd 174 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
15 | mnfnepnf 7745 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
16 | 15 | neii 2284 | . . . . 5 ⊢ ¬ -∞ = +∞ |
17 | eqeq1 2121 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
18 | 16, 17 | mtbiri 647 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
19 | mnfltpnf 9464 | . . . . . . 7 ⊢ -∞ < +∞ | |
20 | breq1 3898 | . . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
21 | 19, 20 | mpbiri 167 | . . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
22 | 21 | necon3bi 2332 | . . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞) |
23 | 22 | necon2bi 2337 | . . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞) |
24 | 18, 23 | 2falsed 674 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
25 | 7, 14, 24 | 3jaoi 1264 | . 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 944 = wceq 1314 ∈ wcel 1463 class class class wbr 3895 ℝcr 7546 +∞cpnf 7721 -∞cmnf 7722 ℝ*cxr 7723 < clt 7724 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-pre-ltirr 7657 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 |
This theorem is referenced by: npnflt 9491 xgepnf 9492 xrmaxiflemlub 10909 |
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