Theorem nn0xnn0d 12244

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 12238	. 2 ⊢ ℕ0 ⊆ ℕ0*
2		nn0xnn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
3	1, 2	sselid 3915	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0*)

Syntax hints:   → wi 4   ∈ wcel 2108  ℕ₀cn0 12163  ℕ₀^*cxnn0 12235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-xnn0 12236

This theorem is referenced by:  xnn0xaddcl  12898  pcxnn0cl  16489  fusgrn0eqdrusgr  27840  cusgrrusgr  27851