Theorem pion 10832

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 10831	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7848	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2109  Oncon0 6332  ωcom 7842  Ncnpi 10797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-ss 3931  df-om 7843  df-ni 10825

This theorem is referenced by:  indpi  10860  nqereu  10882