Theorem piord 10871

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 10869	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7868	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  Ord word 6359  ωcom 7860  Ncnpi 10835

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-ss 3921  df-uni 4872  df-tr 5218  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-om 7861  df-ni 10863

This theorem is used by: (None)