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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 7961 | . . . . . . 7 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2437 | . . . . . 6 ⊢ ¬ +∞ ∈ ℝ |
3 | 2 | intnanr 925 | . . . . 5 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) |
4 | 3 | intnanr 925 | . . . 4 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) |
5 | pnfnemnf 7974 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | neii 2342 | . . . . 5 ⊢ ¬ +∞ = -∞ |
7 | 6 | intnanr 925 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 = +∞) |
8 | 4, 7 | pm3.2ni 808 | . . 3 ⊢ ¬ (((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) |
9 | 2 | intnanr 925 | . . . 4 ⊢ ¬ (+∞ ∈ ℝ ∧ 𝐴 = +∞) |
10 | 6 | intnanr 925 | . . . 4 ⊢ ¬ (+∞ = -∞ ∧ 𝐴 ∈ ℝ) |
11 | 9, 10 | pm3.2ni 808 | . . 3 ⊢ ¬ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 808 | . 2 ⊢ ¬ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
13 | pnfxr 7972 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | ltxr 9732 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
15 | 13, 14 | mpan 422 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ < 𝐴 ↔ ((((+∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ +∞ <ℝ 𝐴) ∨ (+∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((+∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (+∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
16 | 12, 15 | mtbiri 670 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ℝcr 7773 <ℝ cltrr 7778 +∞cpnf 7951 -∞cmnf 7952 ℝ*cxr 7953 < clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 |
This theorem is referenced by: pnfge 9746 xrltnsym 9750 xrlttr 9752 xrltso 9753 xltnegi 9792 xposdif 9839 qbtwnxr 10214 xrmaxiflemab 11210 xrmaxltsup 11221 |
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