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Theorem elxnn0 8708
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 8707 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2154 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 3139 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 7520 . . . 4 +∞ ∈ V
54elsn2 3473 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 714 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 204 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff set class
Syntax hints:  wb 103  wo 664   = wceq 1289  wcel 1438  cun 2995  {csn 3441  +∞cpnf 7498  0cn0 8643  0*cxnn0 8706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-un 4251  ax-cnex 7415
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-pnf 7503  df-xr 7505  df-xnn0 8707
This theorem is referenced by:  xnn0xr  8711  pnf0xnn0  8713  xnn0nemnf  8717  xnn0nnn0pnf  8719
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