Theorem reno 28437

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 28436	. 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 497	1 ⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∃wrex 3058   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356   No csur 27605   <s cslt 27606   |s cscut 27749   1_s c1s 27794   +_s cadds 27929   -u_s cnegs 27988   -_s csubs 27989   /_su cdivs 28156  ℕ_scnns 28274  ℝ_screno 28434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-reno 28435

This theorem is referenced by:  renod  28438