Theorem onno 28247

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 28246	. 2 ⊢ Ons ⊆ No
2	1	sseli 3917	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2114   No csur 27603  On_scons 28243

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-ss 3906  df-ons 28244

This theorem is referenced by:  elons2  28250  ltonold  28253  oncutleft  28255  oncutlt  28256  onnolt  28258  onlts  28259  onles  28260  bdayons  28268  onaddscl  28269  onmulscl  28270  addonbday  28271  onsbnd2  28274  onsfi  28348