Theorem nnssnn0 8291

Description: Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Assertion

Ref	Expression
nnssnn0	⊢ ℕ ⊆ ℕ0

Proof of Theorem nnssnn0

Step	Hyp	Ref	Expression
1		ssun1 3135	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 8289	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtr4i 3032	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 2971   ⊆ wss 2973  {csn 3398  0cc0 6981  ℕcn 8039  ℕ₀cn0 8288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-n0 8289

This theorem is referenced by:  nnnn0  8295  nnnn0d  8341  oddge22np1  10281