MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0ssxnn0 Structured version   Visualization version   GIF version

Theorem nn0ssxnn0 12604
Description: The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
nn0ssxnn0 0 ⊆ ℕ0*

Proof of Theorem nn0ssxnn0
StepHypRef Expression
1 ssun1 4177 . 2 0 ⊆ (ℕ0 ∪ {+∞})
2 df-xnn0 12602 . 2 0* = (ℕ0 ∪ {+∞})
31, 2sseqtrri 4032 1 0 ⊆ ℕ0*
Colors of variables: wff setvar class
Syntax hints:  cun 3948  wss 3950  {csn 4625  +∞cpnf 11293  0cn0 12528  0*cxnn0 12601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-xnn0 12602
This theorem is referenced by:  nn0xnn0  12605  0xnn0  12607  nn0xnn0d  12610
  Copyright terms: Public domain W3C validator