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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 2821 | . . 3 ⊢ +∞ = +∞ | |
2 | 1 | olci 862 | . 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞) |
3 | elxnn0 11963 | . 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞)) | |
4 | 2, 3 | mpbir 233 | 1 ⊢ +∞ ∈ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1533 ∈ wcel 2110 +∞cpnf 10666 ℕ0cn0 11891 ℕ0*cxnn0 11961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-pow 5259 ax-un 7455 ax-cnex 10587 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3497 df-un 3941 df-in 3943 df-ss 3952 df-pw 4541 df-sn 4562 df-uni 4833 df-pnf 10671 df-xnn0 11962 |
This theorem is referenced by: xnn0xaddcl 12622 |
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