Theorem elxnn0 12721

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 ∈ ℕ0 ∨ 𝐴 = +∞))

Proof of Theorem elxnn0

Step	Hyp	Ref	Expression
1		df-xnn0 12720	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4155	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11401	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4627	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 926	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 300	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  +∞cpnf 11379  ℕ₀cn0 12646  ℕ₀^*cxnn0 12719

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-un 7783  ax-cnex 11295

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-uni 4873  df-pnf 11384  df-xnn0 12720

This theorem is used by:  xnn0xr  12724  pnf0xnn0  12726  xnn0nemnf  12730  xnn0nnn0pnf  12732  xnn0n0n1ge2b  13301  xnn0ge0  13303  xnn0lenn0nn0  13415  xnn0xadd0  13417  xnn0xrge0  13677  tayl0  26746  xnn0gt0  33457  xnn0nn0d  33460  fldextrspundgdvdslem  34423