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Theorem elxnn0 11817
 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 11816 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2874 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 4046 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 10540 . . . 4 +∞ ∈ V
54elsn2 4509 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 907 . 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 842   = wceq 1522   ∈ wcel 2081   ∪ cun 3857  {csn 4472  +∞cpnf 10518  ℕ0cn0 11745  ℕ0*cxnn0 11815 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-pow 5157  ax-un 7319  ax-cnex 10439 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-v 3439  df-un 3864  df-in 3866  df-ss 3874  df-pw 4455  df-sn 4473  df-uni 4746  df-pnf 10523  df-xnn0 11816 This theorem is referenced by:  xnn0xr  11820  pnf0xnn0  11822  xnn0nemnf  11826  xnn0nnn0pnf  11828  xnn0n0n1ge2b  12376  xnn0ge0  12378  xnn0lenn0nn0  12488  xnn0xadd0  12490  xnn0xrge0  12741  tayl0  24633  xnn0gt0  30182
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