Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10685 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3128 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | 2 | intnanr 490 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
4 | 3 | intnanr 490 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
5 | pnfnemnf 10699 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | neii 3021 | . . . . 5 ⊢ ¬ +∞ = -∞ |
7 | 6 | intnanr 490 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
8 | 4, 7 | pm3.2ni 877 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
9 | 2 | intnanr 490 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
10 | 6 | intnanr 490 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
11 | 9, 10 | pm3.2ni 877 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 877 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
13 | pnfxr 10698 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | ltxr 12513 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
15 | 13, 14 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
16 | 12, 15 | mtbiri 329 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ℝcr 10539 <ℝ cltrr 10544 +∞cpnf 10675 -∞cmnf 10676 ℝ*cxr 10677 < clt 10678 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-xp 5564 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 |
This theorem is referenced by: pnfge 12528 xrltnsym 12533 xrlttr 12536 qbtwnxr 12596 xltnegi 12612 xmullem2 12661 xrinfmexpnf 12702 xrsupsslem 12703 xrinfmsslem 12704 xrub 12708 supxrpnf 12714 supxrunb1 12715 supxrunb2 12716 xrinf0 12734 lt6abl 19018 pnfnei 21831 metdstri 23462 esumpcvgval 31341 icorempo 34636 iooelexlt 34647 iccpartigtl 43590 |
Copyright terms: Public domain | W3C validator |