Theorem elxnn0 12592

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 12591	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2818	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4145	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11308	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4662	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 910	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 296	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 205   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ∪ cun 3944  {csn 4623  +∞cpnf 11286  ℕ₀cn0 12518  ℕ₀^*cxnn0 12590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-pow 5361  ax-un 7738  ax-cnex 11205

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-un 3951  df-ss 3963  df-pw 4599  df-sn 4624  df-uni 4906  df-pnf 11291  df-xnn0 12591

This theorem is referenced by:  xnn0xr  12595  pnf0xnn0  12597  xnn0nemnf  12601  xnn0nnn0pnf  12603  xnn0n0n1ge2b  13159  xnn0ge0  13161  xnn0lenn0nn0  13272  xnn0xadd0  13274  xnn0xrge0  13531  tayl0  26386  xnn0gt0  32676