Theorem onsno 28234

Description: A surreal ordinal is a surreal. (Contributed by Scott Fenton, 18-Mar-2025.)

Assertion

Ref	Expression
onsno	⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Proof of Theorem onsno

Step	Hyp	Ref	Expression
1		onssno 28233	. 2 ⊢ Ons ⊆ No
2	1	sseli 3928	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2114   No csur 27609  On_scons 28230

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-ss 3917  df-ons 28231

This theorem is referenced by:  elons2  28237  sltonold  28240  onscutleft  28242  onscutlt  28243  onnolt  28245  onslt  28246  onsle  28247  bdayon  28255  onaddscl  28256  onmulscl  28257  addsonbday  28258  onsbnd2  28261  onsfi  28334