Theorem pnf0xnn0 11977

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 2823	. . 3 ⊢ +∞ = +∞
2	1	olci 862	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 11972	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 233	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 843   = wceq 1537   ∈ wcel 2114  +∞cpnf 10674  ℕ₀cn0 11900  ℕ₀^*cxnn0 11970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-pow 5268  ax-un 7463  ax-cnex 10595

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-un 3943  df-in 3945  df-ss 3954  df-pw 4543  df-sn 4570  df-uni 4841  df-pnf 10679  df-xnn0 11971

This theorem is referenced by:  xnn0xaddcl  12631