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Theorem elxnn0 12494
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 12493 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2830 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 4113 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 11215 . . . 4 +∞ ∈ V
54elsn2 4630 . . 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 205  wo 846   = wceq 1542  wcel 2107  cun 3913  {csn 4591  +∞cpnf 11193  0cn0 12420  0*cxnn0 12492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pow 5325  ax-un 7677  ax-cnex 11114
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-in 3922  df-ss 3932  df-pw 4567  df-sn 4592  df-uni 4871  df-pnf 11198  df-xnn0 12493
This theorem is referenced by:  xnn0xr  12497  pnf0xnn0  12499  xnn0nemnf  12503  xnn0nnn0pnf  12505  xnn0n0n1ge2b  13059  xnn0ge0  13061  xnn0lenn0nn0  13171  xnn0xadd0  13173  xnn0xrge0  13430  tayl0  25737  xnn0gt0  31716
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