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Mirrors > Home > ILE Home > Th. List > ngtmnft | GIF version |
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
ngtmnft | ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9556 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renemnf 7807 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
3 | 2 | neneqd 2327 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞) |
4 | mnflt 9562 | . . . . 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 | pnfnemnf 7813 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2319 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 10 | neneqd 2327 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
12 | mnfltpnf 9564 | . . . . . . 7 ⊢ -∞ < +∞ | |
13 | breq2 3928 | . . . . . . 7 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
14 | 12, 13 | mpbiri 167 | . . . . . 6 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
15 | 14 | necon3bi 2356 | . . . . 5 ⊢ (¬ -∞ < 𝐴 → 𝐴 ≠ +∞) |
16 | 15 | necon2bi 2361 | . . . 4 ⊢ (𝐴 = +∞ → ¬ ¬ -∞ < 𝐴) |
17 | 11, 16 | 2falsed 691 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
18 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
19 | mnfxr 7815 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
20 | xrltnr 9559 | . . . . . 6 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ ¬ -∞ < -∞ |
22 | breq2 3928 | . . . . 5 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
23 | 21, 22 | mtbiri 664 | . . . 4 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
24 | 18, 23 | 2thd 174 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
25 | 7, 17, 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 ≠ wne 2306 class class class wbr 3924 ℝcr 7612 +∞cpnf 7790 -∞cmnf 7791 ℝ*cxr 7792 < clt 7793 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 |
This theorem is referenced by: nmnfgt 9594 ge0nemnf 9600 xleaddadd 9663 |
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