Theorem elxnn0 12574

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 12573	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2855	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4107	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11257	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4631	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 925	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 300	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ∪ cun 3903  {csn 4589  +∞cpnf 11235  ℕ₀cn0 12499  ℕ₀^*cxnn0 12572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-un 7732  ax-cnex 11151

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-ss 3922  df-pw 4564  df-sn 4590  df-uni 4873  df-pnf 11240  df-xnn0 12573

This theorem is referenced by:  xnn0xr  12577  pnf0xnn0  12579  xnn0nemnf  12583  xnn0nnn0pnf  12585  xnn0n0n1ge2b  13152  xnn0ge0  13154  xnn0lenn0nn0  13266  xnn0xadd0  13268  xnn0xrge0  13528  tayl0  26525  xnn0gt0  33114  xnn0nn0d  33117  fldextrspundgdvdslem  34070