Theorem onsno 28196

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 28195	. 2 ⊢ Ons ⊆ No
2	1	sseli 3939	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2109   No csur 27584  On_scons 28192

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-ss 3928  df-ons 28193

This theorem is referenced by:  elons2  28199  sltonold  28202  onscutleft  28204  onscutlt  28205  onnolt  28207  onslt  28208  bdayon  28213  onaddscl  28214  onmulscl  28215  onsfi  28287