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Theorem elxnn0 11634
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 11633 . . 3 0* = (ℕ0 ∪ {+∞})
21eleq2i 2884 . 2 (𝐴 ∈ ℕ0*𝐴 ∈ (ℕ0 ∪ {+∞}))
3 elun 3959 . 2 (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 ∈ {+∞}))
4 pnfex 10381 . . . 4 +∞ ∈ V
54elsn2 4412 . . 3 (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
65orbi2i 927 . 2 ((𝐴 ∈ ℕ0𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
72, 3, 63bitri 288 1 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wo 865   = wceq 1637  wcel 2157  cun 3774  {csn 4377  +∞cpnf 10359  0cn0 11562  0*cxnn0 11632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-pow 5042  ax-un 7182  ax-cnex 10280
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rex 3109  df-v 3400  df-un 3781  df-in 3783  df-ss 3790  df-pw 4360  df-sn 4378  df-pr 4380  df-uni 4638  df-pnf 10364  df-xr 10366  df-xnn0 11633
This theorem is referenced by:  xnn0xr  11637  pnf0xnn0  11639  xnn0nemnf  11643  xnn0nnn0pnf  11645  xnn0n0n1ge2b  12184  xnn0ge0  12186  xnn0lenn0nn0  12296  xnn0xadd0  12298  xnn0xrge0  12551  tayl0  24336
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