Theorem nnoni 7884

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 7883	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ On

This proof depends on syntax axioms:   ∈ wcel 2144  Oncon0 6362  ωcom 7877

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-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-om 7878

This theorem is used by:  omopthlem1  8668  omopthlem2  8669  omopthi  8670