Theorem elxnn0 11817

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 11816	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2874	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4046	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 10540	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4509	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 907	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 298	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 207   ∨ wo 842   = wceq 1522   ∈ wcel 2081   ∪ cun 3857  {csn 4472  +∞cpnf 10518  ℕ₀cn0 11745  ℕ₀^*cxnn0 11815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-pow 5157  ax-un 7319  ax-cnex 10439

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-v 3439  df-un 3864  df-in 3866  df-ss 3874  df-pw 4455  df-sn 4473  df-uni 4746  df-pnf 10523  df-xnn0 11816

This theorem is referenced by:  xnn0xr  11820  pnf0xnn0  11822  xnn0nemnf  11826  xnn0nnn0pnf  11828  xnn0n0n1ge2b  12376  xnn0ge0  12378  xnn0lenn0nn0  12488  xnn0xadd0  12490  xnn0xrge0  12741  tayl0  24633  xnn0gt0  30182