Theorem pion 10635

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 10634	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7718	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2106  Oncon0 6266  ωcom 7712  Ncnpi 10600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-om 7713  df-ni 10628

This theorem is referenced by:  indpi  10663  nqereu  10685