Theorem piord 10922

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 10920	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7869	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Ord word 6351  ωcom 7861  Ncnpi 10886

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6355  df-on 6356  df-om 7862  df-ni 10914

This theorem is used by: (None)