Theorem nn0xnn0d 12592

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 12586	. 2 ⊢ ℕ0 ⊆ ℕ0*
2		nn0xnn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
3	1, 2	sselid 3934	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  ℕ₀cn0 12510  ℕ₀^*cxnn0 12583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-xnn0 12584

This theorem is used by:  xnn0xaddcl  13267  pcxnn0cl  16926  fusgrn0eqdrusgr  29931  cusgrrusgr  29942