![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 7831 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2406 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | 2 | intnanr 916 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
4 | 3 | intnanr 916 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
5 | pnfnemnf 7844 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | neii 2311 | . . . . 5 ⊢ ¬ +∞ = -∞ |
7 | 6 | intnanr 916 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
8 | 4, 7 | pm3.2ni 803 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
9 | 2 | intnanr 916 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
10 | 6 | intnanr 916 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
11 | 9, 10 | pm3.2ni 803 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 803 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
13 | pnfxr 7842 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | ltxr 9592 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
15 | 13, 14 | mpan 421 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
16 | 12, 15 | mtbiri 665 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 <ℝ cltrr 7648 +∞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-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 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-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: pnfge 9605 xrltnsym 9609 xrlttr 9611 xrltso 9612 xltnegi 9648 xposdif 9695 qbtwnxr 10066 xrmaxiflemab 11048 xrmaxltsup 11059 |
Copyright terms: Public domain | W3C validator |