Theorem pion 10870

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 10869	. 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 2142  Oncon0 6360  ωcom 7860  Ncnpi 10835

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-ss 3921  df-om 7861  df-ni 10863

This theorem is used by:  indpi  10898  nqereu  10920