Theorem onno 28253

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 28252	. 2 ⊢ Ons ⊆ No
2	1	sseli 3929	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2113   No csur 27609  On_scons 28249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-ss 3918  df-ons 28250

This theorem is referenced by:  elons2  28256  ltonold  28259  oncutleft  28261  oncutlt  28262  onnolt  28264  onlts  28265  onles  28266  bdayons  28274  onaddscl  28275  onmulscl  28276  addonbday  28277  onsbnd2  28280  onsfi  28354