Theorem pnf0xnn0 9380

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 2206	. . 3 ⊢ +∞ = +∞
2	1	olci 734	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 9375	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 146	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 710   = wceq 1373   ∈ wcel 2177  +∞cpnf 8119  ℕ₀cn0 9310  ℕ₀^*cxnn0 9373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-un 4487  ax-cnex 8031

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-uni 3856  df-pnf 8124  df-xr 8126  df-xnn0 9374

This theorem is referenced by:  inftonninf  10604  nninfctlemfo  12431  pcxnn0cl  12703