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Mirrors > Home > MPE Home > Th. List > pnf0xnn0 | Structured version Visualization version GIF version |
Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
pnf0xnn0 | ⊢ +∞ ∈ ℕ0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ +∞ = +∞ | |
2 | 1 | olci 853 | . 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞) |
3 | elxnn0 11779 | . 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞)) | |
4 | 2, 3 | mpbir 223 | 1 ⊢ +∞ ∈ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 834 = wceq 1508 ∈ wcel 2051 +∞cpnf 10469 ℕ0cn0 11705 ℕ0*cxnn0 11777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-sep 5056 ax-pow 5115 ax-un 7277 ax-cnex 10389 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-rex 3087 df-v 3410 df-un 3827 df-in 3829 df-ss 3836 df-pw 4418 df-sn 4436 df-uni 4709 df-pnf 10474 df-xnn0 11778 |
This theorem is referenced by: xnn0xaddcl 12443 |
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