Theorem reno 28587

Description: A surreal real is a surreal number. (Contributed by Scott Fenton, 19-Feb-2026.)

Assertion

Ref	Expression
reno	⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )

Proof of Theorem reno

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreno 28586	. 2 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
2	1	simplbi 500	1 ⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  ∃wrex 3088   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398   No csur 27706   <s clts 27707   |s ccuts 27854   1_s c1s 27901   +_s cadds 28054   -u_s cnegs 28114   -_s csubs 28115   /_su cdivs 28282  ℕ_scnns 28408  ℝ_screno 28584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-reno 28585

This theorem is referenced by:  renod  28588