Theorem elxnn0 12688

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 12687	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4151	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11369	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4622	. . 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 1569   ∈ wcel 2144   ∪ cun 4148  {csn 4584  +∞cpnf 11347  ℕ₀cn0 12613  ℕ₀^*cxnn0 12686

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5327  ax-un 7755  ax-cnex 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-pw 4558  df-sn 4585  df-uni 4867  df-pnf 11352  df-xnn0 12687

This theorem is used by:  xnn0xr  12691  pnf0xnn0  12693  xnn0nemnf  12697  xnn0nnn0pnf  12699  xnn0n0n1ge2b  13268  xnn0ge0  13270  xnn0lenn0nn0  13382  xnn0xadd0  13384  xnn0xrge0  13644  tayl0  26698  xnn0gt0  33411  xnn0nn0d  33414  fldextrspundgdvdslem  34362