Theorem nnssnn0 12602

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 4124	. 2 ⊢ ℕ ⊆ (ℕ ∪ {0})
2		df-n0 12600	. 2 ⊢ ℕ0 = (ℕ ∪ {0})
3	1, 2	sseqtrri 3980	1 ⊢ ℕ ⊆ ℕ0

This proof depends on syntax axioms:   ∪ cun 3897   ⊆ wss 3899  {csn 4584  0cc0 11193  ℕcn 12328  ℕ₀cn0 12599

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-n0 12600

This theorem is used by:  nnnn0  12606  nnnn0d  12660  nthruz  16414  oddge22np1  16512  bitsfzolem  16597  lcmfval  16789  ramub1  17199  ramcl  17200  ply1divex  26448  pserdvlem2  26748  2sqreunnlem1  27769  2sqreunnlem2  27775  fsum2dsub  35229  breprexplemc  35254  breprexpnat  35256  knoppndvlem18  37375  sumcubes  43350  hbtlem5  44114  brfvtrcld  44706  corcltrcl  44724  fourierdlem50  47135  fourierdlem102  47187  fourierdlem114  47199  fmtnoinf  48590  fmtnofac2  48623