Theorem piord 10868

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 10866	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7873	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2150  Ord word 6363  ωcom 7865  Ncnpi 10832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rab 3424  df-v 3464  df-dif 3916  df-ss 3930  df-uni 4878  df-tr 5224  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-ord 6367  df-on 6368  df-om 7866  df-ni 10860

This theorem is referenced by: (None)