Theorem nn0xnn0d 9042

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 9036	. 2 ⊢ ℕ0 ⊆ ℕ0*
2		nn0xnn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
3	1, 2	sseldi 3090	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*)

Syntax hints:   → wi 4   ∈ wcel 1480  ℕ₀cn0 8970  ℕ₀^*cxnn0 9033

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-xnn0 9034

This theorem is referenced by: (None)