Theorem elons 28457

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 6881	. . 3 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
2	1	eqeq1d 2764	. 2 ⊢ (𝑥 = 𝐴 → (( R ‘𝑥) = ∅ ↔ ( R ‘𝐴) = ∅))
3		df-ons 28456	. 2 ⊢ Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
4	2, 3	elrab2 3653	1 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∅c0 4285  ‘cfv 6536   No csur 27815   R cright 28030  On_scons 28455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ons 28456

This theorem is used by:  0ons  28460  1ons  28461  elons2  28462  onleft  28464  ltonold  28465  oncutleft  28467  oncutlt  28468  zcuts0  28612  bdayfinbndlem1  28671