Theorem elxnn0 11310

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 11309	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2696	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 3736	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 10038	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4187	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 541	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 286	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 196   ∨ wo 383   = wceq 1480   ∈ wcel 1992   ∪ cun 3558  {csn 4153  +∞cpnf 10016  ℕ₀cn0 11237  ℕ₀^*cxnn0 11308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-pow 4808  ax-un 6903  ax-cnex 9937

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-v 3193  df-un 3565  df-in 3567  df-ss 3574  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-pnf 10021  df-xr 10023  df-xnn0 11309

This theorem is referenced by:  xnn0xr  11313  pnf0xnn0  11315  xnn0nemnf  11319  xnn0nnn0pnf  11321  xnn0n0n1ge2b  11909  xnn0ge0  11911  xnn0xadd0  12017  xnn0xrge0  12264  tayl0  24015