Theorem elxnn0 12503

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 12502	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2831	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4083	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11189	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4597	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 918	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 298	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ∪ cun 3881  {csn 4555  +∞cpnf 11167  ℕ₀cn0 12428  ℕ₀^*cxnn0 12501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-un 7678  ax-cnex 11085

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-ss 3900  df-pw 4531  df-sn 4556  df-uni 4839  df-pnf 11172  df-xnn0 12502

This theorem is referenced by:  xnn0xr  12506  pnf0xnn0  12508  xnn0nemnf  12512  xnn0nnn0pnf  12514  xnn0n0n1ge2b  13074  xnn0ge0  13076  xnn0lenn0nn0  13188  xnn0xadd0  13190  xnn0xrge0  13450  tayl0  26345  xnn0gt0  32861  xnn0nn0d  32864  fldextrspundgdvdslem  33864