Theorem pnf0xnn0 12726

Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
pnf0xnn0	⊢ +∞ ∈ ℕ0*

Proof of Theorem pnf0xnn0

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ +∞ = +∞
2	1	olci 880	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12721	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ∈ wcel 2147  +∞cpnf 11379  ℕ₀cn0 12646  ℕ₀^*cxnn0 12719

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-un 7783  ax-cnex 11295

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-uni 4873  df-pnf 11384  df-xnn0 12720

This theorem is used by:  xnn0xaddcl  13405  pcxnn0cl  17082