Theorem elxnn0 12512

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 12511	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2828	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 4093	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 11198	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4609	. . 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 3887  {csn 4567  +∞cpnf 11176  ℕ₀cn0 12437  ℕ₀^*cxnn0 12510

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 2708  ax-sep 5231  ax-pow 5307  ax-un 7689  ax-cnex 11094

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-pw 4543  df-sn 4568  df-uni 4851  df-pnf 11181  df-xnn0 12511

This theorem is referenced by:  xnn0xr  12515  pnf0xnn0  12517  xnn0nemnf  12521  xnn0nnn0pnf  12523  xnn0n0n1ge2b  13083  xnn0ge0  13085  xnn0lenn0nn0  13197  xnn0xadd0  13199  xnn0xrge0  13459  tayl0  26327  xnn0gt0  32842  xnn0nn0d  32845  fldextrspundgdvdslem  33824