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| Description: Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| elons | ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . 3 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴)) | |
| 2 | 1 | eqeq1d 2738 | . 2 ⊢ (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅)) | 
| 3 | df-ons 28276 | . 2 ⊢ Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅} | |
| 4 | 2, 3 | elrab2 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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