Theorem reno 28484

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 28483	. 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 496	1 ⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   |s ccuts 27751   1_s c1s 27798   +_s cadds 27951   -u_s cnegs 28011   -_s csubs 28012   /_su cdivs 28179  ℕ_scnns 28305  ℝ_screno 28481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-reno 28482

This theorem is referenced by:  renod  28485