Theorem elreno 28785

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 5107	. . . . 5 ⊢ (𝑦 = 𝐴 → (( -us ‘𝑛) <s 𝑦 ↔ ( -us ‘𝑛) <s 𝐴))
2		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑛 ↔ 𝐴 <s 𝑛))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐴 → ((( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
4	3	rexbidv 3186	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
5		id 23	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
6		oveq1 7422	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 -s ( 1s /su 𝑛)) = (𝐴 -s ( 1s /su 𝑛)))
7	6	eqeq2d 2771	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexbidv 3186	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
9	8	abbidv 2826	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})
10		oveq1 7422	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑛)))
11	10	eqeq2d 2771	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
12	11	rexbidv 3186	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
13	12	abbidv 2826	. . . . 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 7433	. . . 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 2776	. . 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 644	. 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 28784	. 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 3649	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))}))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086   class class class wbr 5103  ‘cfv 6534  (class class class)co 7415   No csur 27905   <s clts 27906   |s ccuts 28053   1_s c1s 28100   +_s cadds 28253   -u_s cnegs 28313   -_s csubs 28314   /_su cdivs 28481  ℕ_scnns 28607  ℝ_screno 28783

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6490  df-fv 6542  df-ov 7418  df-reno 28784

This theorem is used by:  reno  28786  elreno2  28789  renegscl  28792  readdscl  28793  remulscl  28796