Theorem onno 28641

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 28640	. 2 ⊢ Ons ⊆ No
2	1	sseli 3904	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Nocsur 27997  On_scons 28637

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-ons 28638

This theorem is used by:  elons2  28644  ltonold  28647  oncutleft  28649  oncutlt  28650  onnolt  28652  onlts  28653  onles  28654  bdayons  28662  onaddscl  28663  onmulscl  28664  addonbday  28665  onsbnd2  28668  onsfi  28742