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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 12487 | . . 3 ⊢ (𝑁 ∈ ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) | |
2 | pm2.53 849 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ 𝑁 = +∞) → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ ℕ0 → 𝑁 = +∞)) |
4 | 3 | imp 407 | 1 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 +∞cpnf 11186 ℕ0cn0 12413 ℕ0*cxnn0 12485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-pow 5320 ax-un 7672 ax-cnex 11107 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-un 3915 df-in 3917 df-ss 3927 df-pw 4562 df-sn 4587 df-uni 4866 df-pnf 11191 df-xnn0 12486 |
This theorem is referenced by: xnn0xaddcl 13154 xnn0lem1lt 13163 nn0xmulclb 31676 |
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