Theorem elxnn0 12506

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 12505	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2829	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4094	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11192	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4610	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 913	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 297	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3888  {csn 4568  +∞cpnf 11170  ℕ₀cn0 12431  ℕ₀^*cxnn0 12504

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pow 5303  ax-un 7683  ax-cnex 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-pw 4544  df-sn 4569  df-uni 4852  df-pnf 11175  df-xnn0 12505

This theorem is referenced by:  xnn0xr  12509  pnf0xnn0  12511  xnn0nemnf  12515  xnn0nnn0pnf  12517  xnn0n0n1ge2b  13077  xnn0ge0  13079  xnn0lenn0nn0  13191  xnn0xadd0  13193  xnn0xrge0  13453  tayl0  26341  xnn0gt0  32860  xnn0nn0d  32863  fldextrspundgdvdslem  33843