Theorem elxnn0 12596

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 12595	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2857	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4107	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11279	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4633	. . 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 2146   ∪ cun 3904  {csn 4591  +∞cpnf 11257  ℕ₀cn0 12521  ℕ₀^*cxnn0 12594

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-un 7742  ax-cnex 11173

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-pw 4566  df-sn 4592  df-uni 4875  df-pnf 11262  df-xnn0 12595

This theorem is used by:  xnn0xr  12599  pnf0xnn0  12601  xnn0nemnf  12605  xnn0nnn0pnf  12607  xnn0n0n1ge2b  13175  xnn0ge0  13177  xnn0lenn0nn0  13289  xnn0xadd0  13291  xnn0xrge0  13551  tayl0  26578  xnn0gt0  33186  xnn0nn0d  33189  fldextrspundgdvdslem  34136