Theorem piord 10923

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 10921	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7884	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2099  Ord word 6375  ωcom 7876  Ncnpi 10887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3950  df-ss 3964  df-uni 4914  df-tr 5271  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-om 7877  df-ni 10915

This theorem is referenced by: (None)