Theorem nnssnn0 12471

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 4171	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 12469	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 4018	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3945   ⊆ wss 3947  {csn 4627  0cc0 11106  ℕcn 12208  ℕ₀cn0 12468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3952  df-in 3954  df-ss 3964  df-n0 12469

This theorem is referenced by:  nnnn0  12475  nnnn0d  12528  nthruz  16192  oddge22np1  16288  bitsfzolem  16371  lcmfval  16554  ramub1  16957  ramcl  16958  ply1divex  25636  pserdvlem2  25922  2sqreunnlem1  26932  2sqreunnlem2  26938  fsum2dsub  33557  breprexplemc  33582  breprexpnat  33584  knoppndvlem18  35343  hbtlem5  41803  brfvtrcld  42405  corcltrcl  42423  fourierdlem50  44807  fourierdlem102  44859  fourierdlem114  44871  fmtnoinf  46139  fmtnofac2  46172