Theorem onno 28668

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 28667	. 2 ⊢ Ons ⊆ No
2	1	sseli 3909	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Nocsur 28024  On_scons 28664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-ss 3901  df-ons 28665

This theorem is used by:  elons2  28671  ltonold  28674  oncutleft  28676  oncutlt  28677  onnolt  28679  onlts  28680  onles  28681  bdayons  28689  onaddscl  28690  onmulscl  28691  addonbday  28692  onsbnd2  28695  onsfi  28769