Theorem piord 10872

Description: A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
piord	⊢ (𝐴 ∈ N → Ord 𝐴)

Proof of Theorem piord

Step	Hyp	Ref	Expression
1		pinn 10870	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7857	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2098  Ord word 6354  ωcom 7849  Ncnpi 10836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-uni 4901  df-tr 5257  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-om 7850  df-ni 10864

This theorem is referenced by: (None)