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Theorem elons 28277
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.
StepHypRef Expression
1 fveq2 6905 . . 3 (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
21eqeq1d 2738 . 2 (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅))
3 df-ons 28276 . 2 Ons = {𝑥 No ∣ ( R ‘𝑥) = ∅}
42, 3elrab2 3694 1 (𝐴 ∈ Ons ↔ (𝐴 No ∧ ( R ‘𝐴) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  c0 4332  cfv 6560   No csur 27685   R cright 27886  Onscons 28275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ons 28276
This theorem is referenced by:  0ons  28280  1ons  28281  elons2  28282  sltonold  28284  onscutleft  28286
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