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Theorem elxnn0 9227
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 9226 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2244 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 3276 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 7998 . . . 4 +∞ ∈ V
54elsn2 3625 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 762 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 206 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 708   = wceq 1353  wcel 2148  cun 3127  {csn 3591  +∞cpnf 7976  0cn0 9162  0*cxnn0 9225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-un 4430  ax-cnex 7890
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-pnf 7981  df-xr 7983  df-xnn0 9226
This theorem is referenced by:  xnn0xr  9230  pnf0xnn0  9232  xnn0nemnf  9236  xnn0nnn0pnf  9238  xnn0dcle  9786  xnn0letri  9787  xnn0lenn0nn0  9849  xnn0xadd0  9851
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