Theorem piord 10798

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 10796	. 2 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		nnord 7818	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2121  Ord word 6313  ωcom 7810  Ncnpi 10762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-ss 3902  df-uni 4842  df-tr 5183  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-om 7811  df-ni 10790

This theorem is referenced by: (None)