Theorem elxnn0 11963

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 11962	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2904	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4124	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 10688	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4597	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 909	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 299	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 208   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ∪ cun 3933  {csn 4560  +∞cpnf 10666  ℕ₀cn0 11891  ℕ₀^*cxnn0 11961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-pow 5258  ax-un 7455  ax-cnex 10587

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-un 3940  df-in 3942  df-ss 3951  df-pw 4540  df-sn 4561  df-uni 4832  df-pnf 10671  df-xnn0 11962

This theorem is referenced by:  xnn0xr  11966  pnf0xnn0  11968  xnn0nemnf  11972  xnn0nnn0pnf  11974  xnn0n0n1ge2b  12520  xnn0ge0  12522  xnn0lenn0nn0  12632  xnn0xadd0  12634  xnn0xrge0  12885  tayl0  24944  xnn0gt0  30488