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Theorem pnf0xnn0 12026
 Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
pnf0xnn0 +∞ ∈ ℕ0*

Proof of Theorem pnf0xnn0
StepHypRef Expression
1 eqid 2758 . . 3 +∞ = +∞
21olci 863 . 2 (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3 elxnn0 12021 . 2 (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
42, 3mpbir 234 1 +∞ ∈ ℕ0*
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 844   = wceq 1538   ∈ wcel 2111  +∞cpnf 10723  ℕ0cn0 11947  ℕ0*cxnn0 12019 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-pow 5238  ax-un 7465  ax-cnex 10644 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-pw 4499  df-sn 4526  df-uni 4802  df-pnf 10728  df-xnn0 12020 This theorem is referenced by:  xnn0xaddcl  12682
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