Theorem onno 28350

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 28349	. 2 ⊢ Ons ⊆ No
2	1	sseli 3934	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2144   No csur 27706  On_scons 28346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-ss 3923  df-ons 28347

This theorem is referenced by:  elons2  28353  ltonold  28356  oncutleft  28358  oncutlt  28359  onnolt  28361  onlts  28362  onles  28363  bdayons  28371  onaddscl  28372  onmulscl  28373  addonbday  28374  onsbnd2  28377  onsfi  28451