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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 9593 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renemnf 7838 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
3 | 2 | neneqd 2330 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞) |
4 | mnflt 9599 | . . . . 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 | pnfnemnf 7844 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2322 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 10 | neneqd 2330 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
12 | mnfltpnf 9601 | . . . . . . 7 ⊢ -∞ < +∞ | |
13 | breq2 3941 | . . . . . . 7 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
14 | 12, 13 | mpbiri 167 | . . . . . 6 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
15 | 14 | necon3bi 2359 | . . . . 5 ⊢ (¬ -∞ < 𝐴 → 𝐴 ≠ +∞) |
16 | 15 | necon2bi 2364 | . . . 4 ⊢ (𝐴 = +∞ → ¬ ¬ -∞ < 𝐴) |
17 | 11, 16 | 2falsed 692 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
18 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
19 | mnfxr 7846 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
20 | xrltnr 9596 | . . . . . 6 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ ¬ -∞ < -∞ |
22 | breq2 3941 | . . . . 5 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
23 | 21, 22 | mtbiri 665 | . . . 4 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
24 | 18, 23 | 2thd 174 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
25 | 7, 17, 24 | 3jaoi 1282 | . 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 962 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 class class class wbr 3937 ℝcr 7643 +∞cpnf 7821 -∞cmnf 7822 ℝ*cxr 7823 < clt 7824 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 |
This theorem is referenced by: nmnfgt 9631 ge0nemnf 9637 xleaddadd 9700 |
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