Theorem nnssnn0 8184

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 3106	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 8182	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtr4i 2978	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 2915   ⊆ wss 2917  {csn 3375  0cc0 6889  ℕcn 7914  ℕ₀cn0 8181

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-n0 8182

This theorem is referenced by:  nnnn0  8188  nnnn0d  8235