Theorem piord 10840

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 10838	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7856	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2144  Ord word 6347  ωcom 7848  Ncnpi 10804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-ss 3923  df-uni 4868  df-tr 5210  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-on 6352  df-om 7849  df-ni 10832

This theorem is referenced by: (None)