Theorem pion 10793

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 10792	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnon 7816	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∈ wcel 2114  Oncon0 6317  ωcom 7810  Ncnpi 10758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-ss 3907  df-om 7811  df-ni 10786

This theorem is referenced by:  indpi  10821  nqereu  10843