Theorem elons 28154

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 6858	. . 3 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
2	1	eqeq1d 2731	. 2 ⊢ (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅))
3		df-ons 28153	. 2 ⊢ Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
4	2, 3	elrab2 3662	1 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4296  ‘cfv 6511   No csur 27551   R cright 27754  On_scons 28152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ons 28153

This theorem is referenced by:  0ons  28157  1ons  28158  elons2  28159  onsleft  28161  sltonold  28162  onscutleft  28164  onscutlt  28165