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Theorem elxnn0 11957
Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
elxnn0 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))

Proof of Theorem elxnn0
StepHypRef Expression
1 df-xnn0 11956 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2901 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 4122 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 10682 . . . 4 +∞ ∈ V
54elsn2 4594 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 906 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 298 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841   = wceq 1528  wcel 2105  cun 3931  {csn 4557  +∞cpnf 10660  0cn0 11885  0*cxnn0 11955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-pow 5257  ax-un 7450  ax-cnex 10581
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-un 3938  df-in 3940  df-ss 3949  df-pw 4537  df-sn 4558  df-uni 4831  df-pnf 10665  df-xnn0 11956
This theorem is referenced by:  xnn0xr  11960  pnf0xnn0  11962  xnn0nemnf  11966  xnn0nnn0pnf  11968  xnn0n0n1ge2b  12514  xnn0ge0  12516  xnn0lenn0nn0  12626  xnn0xadd0  12628  xnn0xrge0  12879  tayl0  24877  xnn0gt0  30420
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