Theorem elxnn0 11634

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 11633	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
2	1	eleq2i 2884	. 2 ⊢ (𝐴 ∈ ℕ0* ↔ 𝐴 ∈ (ℕ0 ∪ {+∞}))
3		elun 3959	. 2 ⊢ (𝐴 ∈ (ℕ0 ∪ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}))
4		pnfex 10381	. . . 4 ⊢ +∞ ∈ V
5	4	elsn2 4412	. . 3 ⊢ (𝐴 ∈ {+∞} ↔ 𝐴 = +∞)
6	5	orbi2i 927	. 2 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 ∈ {+∞}) ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
7	2, 3, 6	3bitri 288	1 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))

Syntax hints:   ↔ wb 197   ∨ wo 865   = wceq 1637   ∈ wcel 2157   ∪ cun 3774  {csn 4377  +∞cpnf 10359  ℕ₀cn0 11562  ℕ₀^*cxnn0 11632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-pow 5042  ax-un 7182  ax-cnex 10280

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rex 3109  df-v 3400  df-un 3781  df-in 3783  df-ss 3790  df-pw 4360  df-sn 4378  df-pr 4380  df-uni 4638  df-pnf 10364  df-xr 10366  df-xnn0 11633

This theorem is referenced by:  xnn0xr  11637  pnf0xnn0  11639  xnn0nemnf  11643  xnn0nnn0pnf  11645  xnn0n0n1ge2b  12184  xnn0ge0  12186  xnn0lenn0nn0  12296  xnn0xadd0  12298  xnn0xrge0  12551  tayl0  24336