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Mirrors > Home > ILE Home > Th. List > nltmnf | GIF version |
Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
nltmnf | ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 7068 | . . . . . . 7 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2299 | . . . . . 6 ⊢ ¬ -∞ ∈ ℝ |
3 | 2 | intnan 838 | . . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) |
4 | 3 | intnanr 839 | . . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) |
5 | pnfnemnf 8697 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | nesymi 2251 | . . . . 5 ⊢ ¬ -∞ = +∞ |
7 | 6 | intnan 838 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞) |
8 | 4, 7 | pm3.2ni 726 | . . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) |
9 | 6 | intnan 838 | . . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞) |
10 | 2 | intnan 838 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ) |
11 | 9, 10 | pm3.2ni 726 | . . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 726 | . 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))) |
13 | mnfxr 8694 | . . 3 ⊢ -∞ ∈ ℝ* | |
14 | ltxr 8695 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) | |
15 | 13, 14 | mpan2 401 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) |
16 | 12, 15 | mtbiri 600 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 <ℝ cltrr 6893 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 < clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: mnfle 8713 xrltnsym 8714 xrlttr 8716 xrltso 8717 xltnegi 8748 |
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