![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elons | Structured version Visualization version GIF version |
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 6907 | . . 3 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴)) | |
2 | 1 | eqeq1d 2737 | . 2 ⊢ (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅)) |
3 | df-ons 28290 | . 2 ⊢ Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅} | |
4 | 2, 3 | elrab2 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 |
Copyright terms: Public domain | W3C validator |