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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 7807 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2405 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | 2 | intnanr 915 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
4 | 3 | intnanr 915 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
5 | pnfnemnf 7820 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | neii 2310 | . . . . 5 ⊢ ¬ +∞ = -∞ |
7 | 6 | intnanr 915 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
8 | 4, 7 | pm3.2ni 802 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
9 | 2 | intnanr 915 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
10 | 6 | intnanr 915 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
11 | 9, 10 | pm3.2ni 802 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 802 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
13 | pnfxr 7818 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | ltxr 9562 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
15 | 13, 14 | mpan 420 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
16 | 12, 15 | mtbiri 664 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 <ℝ cltrr 7624 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 < clt 7800 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 |
This theorem is referenced by: pnfge 9575 xrltnsym 9579 xrlttr 9581 xrltso 9582 xltnegi 9618 xposdif 9665 qbtwnxr 10035 xrmaxiflemab 11016 xrmaxltsup 11027 |
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