Theorem piord 10965

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 10963	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7885	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Ord word 6361  ωcom 7877  Ncnpi 10929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-uni 4867  df-tr 5212  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-om 7878  df-ni 10957

This theorem is used by: (None)