Theorem onno 28269

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 28268	. 2 ⊢ Ons ⊆ No
2	1	sseli 3913	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2121   No csur 27625  On_scons 28265

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-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-ss 3902  df-ons 28266

This theorem is referenced by:  elons2  28272  ltonold  28275  oncutleft  28277  oncutlt  28278  onnolt  28280  onlts  28281  onles  28282  bdayons  28290  onaddscl  28291  onmulscl  28292  addonbday  28293  onsbnd2  28296  onsfi  28370