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Mirrors > Home > ILE Home > Th. List > pnfnlt | 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 7160 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2341 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | 2 | intnanr 872 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
4 | 3 | intnanr 872 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
5 | pnfnemnf 8851 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | neii 2247 | . . . . 5 ⊢ ¬ +∞ = -∞ |
7 | 6 | intnanr 872 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
8 | 4, 7 | pm3.2ni 759 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
9 | 2 | intnanr 872 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
10 | 6 | intnanr 872 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
11 | 9, 10 | pm3.2ni 759 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 759 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
13 | pnfxr 8846 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | ltxr 8849 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
15 | 13, 14 | mpan 414 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
16 | 12, 15 | mtbiri 632 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 <ℝ cltrr 6985 +∞cpnf 7150 -∞cmnf 7151 ℝ*cxr 7152 < clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 |
This theorem is referenced by: pnfge 8864 xrltnsym 8868 xrlttr 8870 xrltso 8871 xltnegi 8902 qbtwnxr 9266 |
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