Theorem elons 28640

Description: Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025.)

Assertion

Ref	Expression
elons	⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ (R‘𝐴) = ∅))

Proof of Theorem elons

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑥 = 𝐴 → (R‘𝑥) = (R‘𝐴))
2	1	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝐴 → ((R‘𝑥) = ∅ ↔ (R‘𝐴) = ∅))
3		df-ons 28639	. 2 ⊢ Ons = {𝑥 ∈ No ∣ (R‘𝑥) = ∅}
4	2, 3	elrab2 3648	1 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ (R‘𝐴) = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ‘cfv 6538  Nocsur 27998  Rcright 28213  On_scons 28638

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-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ons 28639

This theorem is used by:  0ons  28643  1ons  28644  elons2  28645  onleft  28647  ltonold  28648  oncutleft  28650  oncutlt  28651  zcuts0  28795  bdayfinbndlem1  28854