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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 11242 | . . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | pm2.53 387 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) | |
3 | 1, 2 | sylbi 206 | . 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) |
4 | 3 | imp 444 | 1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 +∞cpnf 9950 ℕ0cn0 11169 ℕ0*cxnn0 11240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-pow 4769 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 df-pnf 9955 df-xr 9957 df-xnn0 11241 |
This theorem is referenced by: xnn0xaddcl 11940 |
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