Theorem onsno 28163

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 28162	. 2 ⊢ Ons ⊆ No
2	1	sseli 3950	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2109   No csur 27558  On_scons 28159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-ss 3939  df-ons 28160

This theorem is referenced by:  elons2  28166  sltonold  28169  onscutleft  28171  onscutlt  28172  onnolt  28174  onslt  28175  bdayon  28180  onaddscl  28181  onmulscl  28182  onsfi  28254