Theorem onno 28314

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 28313	. 2 ⊢ Ons ⊆ No
2	1	sseli 3923	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∈ wcel 2132   No csur 27670  On_scons 28310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-ss 3912  df-ons 28311

This theorem is referenced by:  elons2  28317  ltonold  28320  oncutleft  28322  oncutlt  28323  onnolt  28325  onlts  28326  onles  28327  bdayons  28335  onaddscl  28336  onmulscl  28337  addonbday  28338  onsbnd2  28341  onsfi  28415