Theorem onno 28560

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 28559	. 2 ⊢ Ons ⊆ No
2	1	sseli 3927	1 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   No csur 27916  On_scons 28556

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3916  df-ons 28557

This theorem is used by:  elons2  28563  ltonold  28566  oncutleft  28568  oncutlt  28569  onnolt  28571  onlts  28572  onles  28573  bdayons  28581  onaddscl  28582  onmulscl  28583  addonbday  28584  onsbnd2  28587  onsfi  28661