Theorem onno 28414

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 28413	. 2 ⊢ Ons ⊆ No
2	1	sseli 3939	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2149   No csur 27770  On_scons 28410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-ss 3928  df-ons 28411

This theorem is referenced by:  elons2  28417  ltonold  28420  oncutleft  28422  oncutlt  28423  onnolt  28425  onlts  28426  onles  28427  bdayons  28435  onaddscl  28436  onmulscl  28437  addonbday  28438  onsbnd2  28441  onsfi  28515