Theorem elxnn0 12474

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 12473	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4103	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11183	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4620	. . 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 1541   ∈ wcel 2113   ∪ cun 3897  {csn 4578  +∞cpnf 11161  ℕ₀cn0 12399  ℕ₀^*cxnn0 12472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-pow 5308  ax-un 7678  ax-cnex 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-ss 3916  df-pw 4554  df-sn 4579  df-uni 4862  df-pnf 11166  df-xnn0 12473

This theorem is referenced by:  xnn0xr  12477  pnf0xnn0  12479  xnn0nemnf  12483  xnn0nnn0pnf  12485  xnn0n0n1ge2b  13044  xnn0ge0  13046  xnn0lenn0nn0  13158  xnn0xadd0  13160  xnn0xrge0  13420  tayl0  26323  xnn0gt0  32798  xnn0nn0d  32801  fldextrspundgdvdslem  33786