Theorem onsno 28192

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 28191	. 2 ⊢ Ons ⊆ No
2	1	sseli 3925	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2111   No csur 27578  On_scons 28188

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-ss 3914  df-ons 28189

This theorem is referenced by:  elons2  28195  sltonold  28198  onscutleft  28200  onscutlt  28201  onnolt  28203  onslt  28204  bdayon  28209  onaddscl  28210  onmulscl  28211  onsfi  28283