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Theorem elxnn0 12599
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 12598 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2831 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 4163 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 11312 . . . 4 +∞ ∈ V
54elsn2 4670 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 912 . 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 847   = wceq 1537  wcel 2106  cun 3961  {csn 4631  +∞cpnf 11290  0cn0 12524  0*cxnn0 12597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-un 7754  ax-cnex 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-uni 4913  df-pnf 11295  df-xnn0 12598
This theorem is referenced by:  xnn0xr  12602  pnf0xnn0  12604  xnn0nemnf  12608  xnn0nnn0pnf  12610  xnn0n0n1ge2b  13171  xnn0ge0  13173  xnn0lenn0nn0  13284  xnn0xadd0  13286  xnn0xrge0  13543  tayl0  26418  xnn0gt0  32780
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