Theorem nn0xnn0 12688

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

Assertion

Ref	Expression
nn0xnn0	⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)

Proof of Theorem nn0xnn0

Step	Hyp	Ref	Expression
1		nn0ssxnn0 12687	. 2 ⊢ ℕ0 ⊆ ℕ0*
2	1	sseli 3905	1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ℕ₀cn0 12611  ℕ₀^*cxnn0 12684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-xnn0 12685

This theorem is used by:  xnn0xadd0  13382  wlk1ewlk  30227  frgrregorufrg  30934  usgrcyclgt2v  35988