Theorem reno 28694

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 28693	. 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 501	1 ⊢ (𝐴 ∈ ℝs → 𝐴 ∈ No )

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410   No csur 27813   <s clts 27814   |s ccuts 27961   1_s c1s 28008   +_s cadds 28161   -u_s cnegs 28221   -_s csubs 28222   /_su cdivs 28389  ℕ_scnns 28515  ℝ_screno 28691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-reno 28692

This theorem is used by:  renod  28695