Theorem elxnn0 12043

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 12042	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2824	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4037	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 10765	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4552	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 912	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 300	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 209   ∨ wo 846   = wceq 1542   ∈ wcel 2113   ∪ cun 3839  {csn 4513  +∞cpnf 10743  ℕ₀cn0 11969  ℕ₀^*cxnn0 12041

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5164  ax-pow 5229  ax-un 7473  ax-cnex 10664

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-un 3846  df-in 3848  df-ss 3858  df-pw 4487  df-sn 4514  df-uni 4794  df-pnf 10748  df-xnn0 12042

This theorem is referenced by:  xnn0xr  12046  pnf0xnn0  12048  xnn0nemnf  12052  xnn0nnn0pnf  12054  xnn0n0n1ge2b  12602  xnn0ge0  12604  xnn0lenn0nn0  12714  xnn0xadd0  12716  xnn0xrge0  12973  tayl0  25101  xnn0gt0  30659