Theorem nn0xnn0 9424

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

Assertion

Ref	Expression
nn0xnn0	⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)

Proof of Theorem nn0xnn0

Step	Hyp	Ref	Expression
1		nn0ssxnn0 9423	. 2 ⊢ ℕ0 ⊆ ℕ0*
2	1	sseli 3220	1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)

Syntax hints:   → wi 4   ∈ wcel 2200  ℕ₀cn0 9357  ℕ₀^*cxnn0 9420

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-xnn0 9421

This theorem is referenced by:  xnn0xadd0  10051  1tonninf  10650