Theorem nn0ssxnn0 12687

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 4175	. 2 ⊢ ℕ0 ⊆ (ℕ0 ∪ {+∞})
2		df-xnn0 12685	. 2 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
3	1, 2	sseqtrri 3958	1 ⊢ ℕ0 ⊆ ℕ0*

This proof depends on syntax axioms:   ⊆ wss 3896   ∪ cun 4148  {csn 4584  +∞cpnf 11345  ℕ₀cn0 12611  ℕ₀^*cxnn0 12684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-xnn0 12685

This theorem is used by:  nn0xnn0  12688  0xnn0  12690  nn0xnn0d  12693