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Theorem elxnn0 12601
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 12600 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2833 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 4153 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 11314 . . . 4 +∞ ∈ V
54elsn2 4665 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 913 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 297 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848   = wceq 1540  wcel 2108  cun 3949  {csn 4626  +∞cpnf 11292  0cn0 12526  0*cxnn0 12599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pow 5365  ax-un 7755  ax-cnex 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-uni 4908  df-pnf 11297  df-xnn0 12600
This theorem is referenced by:  xnn0xr  12604  pnf0xnn0  12606  xnn0nemnf  12610  xnn0nnn0pnf  12612  xnn0n0n1ge2b  13174  xnn0ge0  13176  xnn0lenn0nn0  13287  xnn0xadd0  13289  xnn0xrge0  13546  tayl0  26403  xnn0gt0  32773  fldextrspundgdvdslem  33730
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