Theorem nnoni 7872

Description: A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)

Hypothesis

Ref	Expression
nnoni.1	⊢ 𝐴 ∈ ω

Assertion

Ref	Expression
nnoni	⊢ 𝐴 ∈ On

Proof of Theorem nnoni

Step	Hyp	Ref	Expression
1		nnoni.1	. 2 ⊢ 𝐴 ∈ ω
2		nnon 7871	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ On

This proof depends on syntax axioms:   ∈ wcel 2145  Oncon0 6361  ωcom 7865

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 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-ss 3919  df-om 7866

This theorem is used by:  omopthlem1  8650  omopthlem2  8651  omopthi  8652