Theorem elxnn0 12576

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 12575	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4128	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11288	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4641	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 912	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 297	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ∪ cun 3924  {csn 4601  +∞cpnf 11266  ℕ₀cn0 12501  ℕ₀^*cxnn0 12574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-pow 5335  ax-un 7729  ax-cnex 11185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-pw 4577  df-sn 4602  df-uni 4884  df-pnf 11271  df-xnn0 12575

This theorem is referenced by:  xnn0xr  12579  pnf0xnn0  12581  xnn0nemnf  12585  xnn0nnn0pnf  12587  xnn0n0n1ge2b  13148  xnn0ge0  13150  xnn0lenn0nn0  13261  xnn0xadd0  13263  xnn0xrge0  13523  tayl0  26321  xnn0gt0  32746  xnn0nn0d  32749  fldextrspundgdvdslem  33721