Theorem elxnn0 12496

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 12495	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2824	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4113	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11217	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4630	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 911	. 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 1541   ∈ wcel 2106   ∪ cun 3911  {csn 4591  +∞cpnf 11195  ℕ₀cn0 12422  ℕ₀^*cxnn0 12494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-pow 5325  ax-un 7677  ax-cnex 11116

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-un 3918  df-in 3920  df-ss 3930  df-pw 4567  df-sn 4592  df-uni 4871  df-pnf 11200  df-xnn0 12495

This theorem is referenced by:  xnn0xr  12499  pnf0xnn0  12501  xnn0nemnf  12505  xnn0nnn0pnf  12507  xnn0n0n1ge2b  13061  xnn0ge0  13063  xnn0lenn0nn0  13174  xnn0xadd0  13176  xnn0xrge0  13433  tayl0  25758  xnn0gt0  31742