![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9776 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renepnf 8005 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | 2 | neneqd 2368 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞) |
4 | ltpnf 9780 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | notnot 629 | . . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞) |
7 | 3, 6 | 2falsed 702 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
8 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
9 | pnfxr 8010 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9779 | . . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ +∞ < +∞ |
12 | breq1 4007 | . . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 675 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 8, 13 | 2thd 175 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
15 | mnfnepnf 8013 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
16 | 15 | neii 2349 | . . . . 5 ⊢ ¬ -∞ = +∞ |
17 | eqeq1 2184 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞)) | |
18 | 16, 17 | mtbiri 675 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞) |
19 | mnfltpnf 9785 | . . . . . . 7 ⊢ -∞ < +∞ | |
20 | breq1 4007 | . . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞)) | |
21 | 19, 20 | mpbiri 168 | . . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞) |
22 | 21 | necon3bi 2397 | . . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞) |
23 | 22 | necon2bi 2402 | . . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞) |
24 | 18, 23 | 2falsed 702 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
25 | 7, 14, 24 | 3jaoi 1303 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
26 | 1, 25 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 class class class wbr 4004 ℝcr 7810 +∞cpnf 7989 -∞cmnf 7990 ℝ*cxr 7991 < clt 7992 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-pre-ltirr 7923 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-xp 4633 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 |
This theorem is referenced by: npnflt 9815 xgepnf 9816 xrmaxiflemlub 11256 |
Copyright terms: Public domain | W3C validator |