Theorem elreno 28502

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 5077	. . . . 5 ⊢ (𝑦 = 𝐴 → (( -us ‘𝑛) <s 𝑦 ↔ ( -us ‘𝑛) <s 𝐴))
2		breq1 5076	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑛 ↔ 𝐴 <s 𝑛))
3	1, 2	anbi12d 638	. . . 4 ⊢ (𝑦 = 𝐴 → ((( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
4	3	rexbidv 3163	. . 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 7364	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 -s ( 1s /su 𝑛)) = (𝐴 -s ( 1s /su 𝑛)))
7	6	eqeq2d 2750	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexbidv 3163	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
9	8	abbidv 2805	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})
10		oveq1 7364	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑛)))
11	10	eqeq2d 2750	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
12	11	rexbidv 3163	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
13	12	abbidv 2805	. . . . 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 7375	. . . 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 2755	. . 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 638	. 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 28501	. 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 3632	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃wrex 3063   class class class wbr 5073  ‘cfv 6486  (class class class)co 7357   No csur 27622   <s clts 27623   |s ccuts 27770   1_s c1s 27817   +_s cadds 27970   -u_s cnegs 28030   -_s csubs 28031   /_su cdivs 28198  ℕ_scnns 28324  ℝ_screno 28500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-iota 6442  df-fv 6494  df-ov 7360  df-reno 28501

This theorem is referenced by:  reno  28503  elreno2  28506  renegscl  28509  readdscl  28510  remulscl  28513