Theorem elxnn0 12459

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 12458	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2820	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4104	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11168	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4617	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 912	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 297	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∪ cun 3901  {csn 4577  +∞cpnf 11146  ℕ₀cn0 12384  ℕ₀^*cxnn0 12457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pow 5304  ax-un 7671  ax-cnex 11065

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-un 3908  df-ss 3920  df-pw 4553  df-sn 4578  df-uni 4859  df-pnf 11151  df-xnn0 12458

This theorem is referenced by:  xnn0xr  12462  pnf0xnn0  12464  xnn0nemnf  12468  xnn0nnn0pnf  12470  xnn0n0n1ge2b  13034  xnn0ge0  13036  xnn0lenn0nn0  13147  xnn0xadd0  13149  xnn0xrge0  13409  tayl0  26267  xnn0gt0  32712  xnn0nn0d  32715  fldextrspundgdvdslem  33647