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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 12147 | . . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | pm2.53 851 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) |
4 | 3 | imp 410 | 1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 +∞cpnf 10847 ℕ0cn0 12073 ℕ0*cxnn0 12145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-pow 5247 ax-un 7512 ax-cnex 10768 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 df-un 3862 df-in 3864 df-ss 3874 df-pw 4505 df-sn 4532 df-uni 4810 df-pnf 10852 df-xnn0 12146 |
This theorem is referenced by: xnn0xaddcl 12808 xnn0lem1lt 12817 nn0xmulclb 30786 |
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