Theorem nnssnn0 11333

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 3809	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 11331	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtr4i 3671	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3605   ⊆ wss 3607  {csn 4210  0cc0 9974  ℕcn 11058  ℕ₀cn0 11330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-n0 11331

This theorem is referenced by:  nnnn0  11337  nnnn0d  11389  nthruz  15026  oddge22np1  15120  bitsfzolem  15203  lcmfval  15381  ramub1  15779  ramcl  15780  ply1divex  23941  pserdvlem2  24227  fsum2dsub  30813  breprexplemc  30838  breprexpnat  30840  knoppndvlem18  32645  hbtlem5  38015  brfvtrcld  38330  corcltrcl  38348  fourierdlem50  40691  fourierdlem102  40743  fourierdlem114  40755  fmtnoinf  41773  fmtnofac2  41806