Theorem pion 10964

Description: A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
pion	⊢ (𝐴 ∈ N → 𝐴 ∈ On)

Proof of Theorem pion

Step	Hyp	Ref	Expression
1		pinn 10963	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7883	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Oncon0 6362  ωcom 7877  Ncnpi 10929

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-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-om 7878  df-ni 10957

This theorem is used by:  indpi  10992  nqereu  11014