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Theorem pnf0xnn0 11408
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 2651 . . 3 +∞ = +∞
21olci 405 . 2 (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3 elxnn0 11403 . 2 (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
42, 3mpbir 221 1 +∞ ∈ ℕ0*
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1523  wcel 2030  +∞cpnf 10109  0cn0 11330  0*cxnn0 11401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-pnf 10114  df-xr 10116  df-xnn0 11402
This theorem is referenced by:  xnn0xaddcl  12104
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