Theorem elxnn0 12556

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 12555	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2854	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4106	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11235	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4624	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 923	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 299	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 208   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ∪ cun 3902  {csn 4582  +∞cpnf 11213  ℕ₀cn0 12481  ℕ₀^*cxnn0 12554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-un 7718  ax-cnex 11129

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4557  df-sn 4583  df-uni 4866  df-pnf 11218  df-xnn0 12555

This theorem is referenced by:  xnn0xr  12559  pnf0xnn0  12561  xnn0nemnf  12565  xnn0nnn0pnf  12567  xnn0n0n1ge2b  13134  xnn0ge0  13136  xnn0lenn0nn0  13248  xnn0xadd0  13250  xnn0xrge0  13510  tayl0  26425  xnn0gt0  32971  xnn0nn0d  32974  fldextrspundgdvdslem  33977