Theorem elons 28669

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 6908	. . 3 ⊢ (𝑥 = 𝐴 → (R‘𝑥) = (R‘𝐴))
2	1	eqeq1d 2765	. 2 ⊢ (𝑥 = 𝐴 → ((R‘𝑥) = ∅ ↔ (R‘𝐴) = ∅))
3		df-ons 28668	. 2 ⊢ Ons = {𝑥 ∈ No ∣ (R‘𝑥) = ∅}
4	2, 3	elrab2 3652	1 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ (R‘𝐴) = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ‘cfv 6558  Nocsur 28027  Rcright 28242  On_scons 28667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-ons 28668

This theorem is used by:  0ons  28672  1ons  28673  elons2  28674  onleft  28676  ltonold  28677  oncutleft  28679  oncutlt  28680  zcuts0  28824  bdayfinbndlem1  28883