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| Mirrors > Home > MPE Home > Th. List > xnn0nnn0pnf | Structured version Visualization version GIF version | ||
| Description: An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
| Ref | Expression |
|---|---|
| xnn0nnn0pnf | ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elxnn0 12601 | . . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
| 2 | pm2.53 852 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) | |
| 3 | 1, 2 | sylbi 217 | . 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) |
| 4 | 3 | imp 406 | 1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 +∞cpnf 11292 ℕ0cn0 12526 ℕ0*cxnn0 12599 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pow 5365 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-uni 4908 df-pnf 11297 df-xnn0 12600 |
| This theorem is referenced by: xnn0xaddcl 13277 xnn0lem1lt 13286 nn0xmulclb 32775 |
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