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Theorem elons 28291
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 6907 . . 3 (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
21eqeq1d 2737 . 2 (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅))
3 df-ons 28290 . 2 Ons = {𝑥 No ∣ ( R ‘𝑥) = ∅}
42, 3elrab2 3698 1 (𝐴 ∈ Ons ↔ (𝐴 No ∧ ( R ‘𝐴) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  c0 4339  cfv 6563   No csur 27699   R cright 27900  Onscons 28289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ons 28290
This theorem is referenced by:  0ons  28294  1ons  28295  elons2  28296  sltonold  28298  onscutleft  28300
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