Theorem piord 10989

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 10987	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7906	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Ord word 6376  ωcom 7898  Ncnpi 10953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6380  df-on 6381  df-om 7899  df-ni 10981

This theorem is used by: (None)