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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 2726 | . . 3 ⊢ +∞ = +∞ | |
2 | 1 | olci 864 | . 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞) |
3 | elxnn0 12598 | . 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞)) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ +∞ ∈ ℕ0* |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1534 ∈ wcel 2099 +∞cpnf 11295 ℕ0cn0 12524 ℕ0*cxnn0 12596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-pow 5369 ax-un 7746 ax-cnex 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3952 df-ss 3964 df-pw 4609 df-sn 4634 df-uni 4914 df-pnf 11300 df-xnn0 12597 |
This theorem is referenced by: xnn0xaddcl 13268 pcxnn0cl 16862 |
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