Theorem pion 10891

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 10890	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7871	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Oncon0 6361  ωcom 7865  Ncnpi 10856

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-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-ss 3919  df-om 7866  df-ni 10884

This theorem is used by:  indpi  10919  nqereu  10941