Theorem onno 28459

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 28458	. 2 ⊢ Ons ⊆ No
2	1	sseli 3932	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   No csur 27815  On_scons 28455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-ss 3921  df-ons 28456

This theorem is used by:  elons2  28462  ltonold  28465  oncutleft  28467  oncutlt  28468  onnolt  28470  onlts  28471  onles  28472  bdayons  28480  onaddscl  28481  onmulscl  28482  addonbday  28483  onsbnd2  28486  onsfi  28560