Theorem nn0ssxnn0 8739

Description: The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
nn0ssxnn0	⊢ ℕ0 ⊆ ℕ0*

Proof of Theorem nn0ssxnn0

Step	Hyp	Ref	Expression
1		ssun1 3163	. 2 ⊢ ℕ0 ⊆ (ℕ0 ∪ {+∞})
2		df-xnn0 8737	. 2 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
3	1, 2	sseqtr4i 3059	1 ⊢ ℕ0 ⊆ ℕ0*

Syntax hints:   ∪ cun 2997   ⊆ wss 2999  {csn 3446  +∞cpnf 7519  ℕ₀cn0 8673  ℕ₀^*cxnn0 8736

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-xnn0 8737

This theorem is referenced by:  nn0xnn0  8740  0xnn0  8742  nn0xnn0d  8745