Theorem pion 10765

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 10764	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7797	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2111  Oncon0 6301  ωcom 7791  Ncnpi 10730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-ss 3914  df-om 7792  df-ni 10758

This theorem is referenced by:  indpi  10793  nqereu  10815