Theorem elxnn0 12578

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 12577	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4115	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11261	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4636	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 925	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 300	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ∪ cun 3911  {csn 4594  +∞cpnf 11239  ℕ₀cn0 12503  ℕ₀^*cxnn0 12576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-un 7733  ax-cnex 11155

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-pw 4569  df-sn 4595  df-uni 4877  df-pnf 11244  df-xnn0 12577

This theorem is referenced by:  xnn0xr  12581  pnf0xnn0  12583  xnn0nemnf  12587  xnn0nnn0pnf  12589  xnn0n0n1ge2b  13156  xnn0ge0  13158  xnn0lenn0nn0  13270  xnn0xadd0  13272  xnn0xrge0  13532  tayl0  26490  xnn0gt0  33054  xnn0nn0d  33057  fldextrspundgdvdslem  34014