Theorem onno 28268

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

Assertion

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

Proof of Theorem onno

Step	Hyp	Ref	Expression
1		onssno 28267	. 2 ⊢ Ons ⊆ No
2	1	sseli 3931	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2114   No csur 27624  On_scons 28264

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-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-ss 3920  df-ons 28265

This theorem is referenced by:  elons2  28271  ltonold  28274  oncutleft  28276  oncutlt  28277  onnolt  28279  onlts  28280  onles  28281  bdayons  28289  onaddscl  28290  onmulscl  28291  addonbday  28292  onsbnd2  28295  onsfi  28369