ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnssnn0 GIF version

Theorem nnssnn0 9109
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
StepHypRef Expression
1 ssun1 3281 . 2 ℕ ⊆ (ℕ ∪ {0})
2 df-n0 9107 . 2 0 = (ℕ ∪ {0})
31, 2sseqtrri 3173 1 ℕ ⊆ ℕ0
Colors of variables: wff set class
Syntax hints:  cun 3110  wss 3112  {csn 3571  0cc0 7745  cn 8849  0cn0 9106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-n0 9107
This theorem is referenced by:  nnnn0  9113  nnnn0d  9159  expcnv  11435  oddge22np1  11807
  Copyright terms: Public domain W3C validator