Theorem pion 10935

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 10934	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7866	. 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 6351  ωcom 7860  Ncnpi 10900

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 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-ss 3915  df-om 7861  df-ni 10928

This theorem is used by:  indpi  10963  nqereu  10985