Theorem reno 28908

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 28907	. 2 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘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 502	1 ⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   <s clts 28028   |s ccuts 28175   1_s c1s 28222   +_s cadds 28375   -_scnegs 28435   −_s csubs 28436   /_su cdivs 28603  ℕ_scnns 28729  ℝ_screno 28905

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-rex 3091  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-ov 7446  df-reno 28906

This theorem is used by:  renod  28909