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Theorem elxnn0 11557
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 11556 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2831 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 3896 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 10285 . . . 4 +∞ ∈ V
54elsn2 4356 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 542 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 286 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1632  wcel 2139  cun 3713  {csn 4321  +∞cpnf 10263  0cn0 11484  0*cxnn0 11555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-pow 4992  ax-un 7114  ax-cnex 10184
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-pnf 10268  df-xr 10270  df-xnn0 11556
This theorem is referenced by:  xnn0xr  11560  pnf0xnn0  11562  xnn0nemnf  11566  xnn0nnn0pnf  11568  xnn0n0n1ge2b  12158  xnn0ge0  12160  xnn0lenn0nn0  12268  xnn0xadd0  12270  xnn0xrge0  12518  tayl0  24315
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