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 7820	. 2 ⊢ (𝐴 ∈ ω → Ord 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ N → Ord 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114  Ord word 6318  ωcom 7812  Ncnpi 10762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-ss 3907  df-uni 4852  df-tr 5194  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-ord 6322  df-on 6323  df-om 7813  df-ni 10790

This theorem is referenced by: (None)