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| Mirrors > Home > MPE Home > Th. List > pnfnlt | Structured version Visualization version GIF version | ||
| Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.) |
| Ref | Expression |
|---|---|
| pnfnlt | ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pnfnre 11302 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
| 2 | 1 | neli 3048 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
| 3 | 2 | intnanr 487 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
| 4 | 3 | intnanr 487 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
| 5 | pnfnemnf 11316 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
| 6 | 5 | neii 2942 | . . . . 5 ⊢ ¬ +∞ = -∞ |
| 7 | 6 | intnanr 487 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
| 8 | 4, 7 | pm3.2ni 881 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
| 9 | 2 | intnanr 487 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
| 10 | 6 | intnanr 487 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
| 11 | 9, 10 | pm3.2ni 881 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
| 12 | 8, 11 | pm3.2ni 881 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
| 13 | pnfxr 11315 | . . 3 ⊢ +∞ ∈ ℝ* | |
| 14 | ltxr 13157 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
| 15 | 13, 14 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
| 16 | 12, 15 | mtbiri 327 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 <ℝ cltrr 11159 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: pnfge 13172 xrltnsym 13179 xrlttr 13182 qbtwnxr 13242 xltnegi 13258 xmullem2 13307 xrinfmexpnf 13348 xrsupsslem 13349 xrinfmsslem 13350 xrub 13354 supxrpnf 13360 supxrunb1 13361 supxrunb2 13362 xrinf0 13380 lt6abl 19913 pnfnei 23228 metdstri 24873 esumpcvgval 34079 icorempo 37352 iooelexlt 37363 iccpartigtl 47410 |
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