Theorem nnssnn0 11295

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 3776	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 11293	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtr4i 3638	1 ⊢ ℕ ⊆ ℕ0

Syntax hints:   ∪ cun 3572   ⊆ wss 3574  {csn 4177  0cc0 9936  ℕcn 11020  ℕ₀cn0 11292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-n0 11293

This theorem is referenced by:  nnnn0  11299  nnnn0d  11351  nthruz  14982  oddge22np1  15073  bitsfzolem  15156  lcmfval  15334  ramub1  15732  ramcl  15733  ply1divex  23896  pserdvlem2  24182  fsum2dsub  30685  breprexplemc  30710  breprexpnat  30712  knoppndvlem18  32520  hbtlem5  37698  brfvtrcld  38013  corcltrcl  38031  fourierdlem50  40373  fourierdlem102  40425  fourierdlem114  40437  fmtnoinf  41448  fmtnofac2  41481