Theorem nn0xnn0d 9186

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 9180	. 2 ⊢ ℕ0 ⊆ ℕ0*
2		nn0xnn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
3	1, 2	sselid 3140	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*)

Syntax hints:   → wi 4   ∈ wcel 2136  ℕ₀cn0 9114  ℕ₀^*cxnn0 9177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-xnn0 9178

This theorem is referenced by:  pcxnn0cl  12242