Theorem nn0xnn0d 9437

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

Hypothesis

Ref	Expression
nn0xnn0d.1	⊢ (𝜑 → 𝐴 ∈ ℕ0)

Assertion

Ref	Expression
nn0xnn0d	⊢ (𝜑 → 𝐴 ∈ ℕ0*)

Proof of Theorem nn0xnn0d

Step	Hyp	Ref	Expression
1		nn0ssxnn0 9431	. 2 ⊢ ℕ0 ⊆ ℕ0*
2		nn0xnn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
3	1, 2	sselid 3222	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*)

Syntax hints:   → wi 4   ∈ wcel 2200  ℕ₀cn0 9365  ℕ₀^*cxnn0 9428

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 9429

This theorem is referenced by:  pcxnn0cl  12828