Theorem elreno 28295

Description: Membership in the set of surreal reals. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
elreno	⊢ (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))}))))

Distinct variable group:   𝑥,𝐴,𝑛

Proof of Theorem elreno

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5153	. . . . 5 ⊢ (𝑦 = 𝐴 → (( -us ‘𝑛) <s 𝑦 ↔ ( -us ‘𝑛) <s 𝐴))
2		breq1 5152	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑛 ↔ 𝐴 <s 𝑛))
3	1, 2	anbi12d 630	. . . 4 ⊢ (𝑦 = 𝐴 → ((( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
4	3	rexbidv 3168	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
5		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
6		oveq1 7426	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 -s ( 1s /su 𝑛)) = (𝐴 -s ( 1s /su 𝑛)))
7	6	eqeq2d 2736	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexbidv 3168	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
9	8	abbidv 2794	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})
10		oveq1 7426	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑛)))
11	10	eqeq2d 2736	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
12	11	rexbidv 3168	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
13	12	abbidv 2794	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
14	9, 13	oveq12d 7437	. . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}) = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
15	5, 14	eqeq12d 2741	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}) ↔ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})))
16	4, 15	anbi12d 630	. 2 ⊢ (𝑦 = 𝐴 → ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ∧ 𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))})) ↔ (∃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))}))))
17		df-reno 28294	. 2 ⊢ ℝs = {𝑦 ∈ No ∣ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ∧ 𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}))}
18	16, 17	elrab2 3682	1 ⊢ (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))}))))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3059   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419   No csur 27618   <s cslt 27619   |s cscut 27761   1_s c1s 27802   +_s cadds 27922   -u_s cnegs 27978   -_s csubs 27979   /_su cdivs 28137  ℕ_scnns 28236  ℝ_screno 28293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-reno 28294

This theorem is referenced by:  0reno  28297  renegscl  28298  readdscl  28299  remulscl  28302