Theorem elreno 28504

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 5104	. . . . 5 ⊢ (𝑦 = 𝐴 → (( -us ‘𝑛) <s 𝑦 ↔ ( -us ‘𝑛) <s 𝐴))
2		breq1 5103	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑛 ↔ 𝐴 <s 𝑛))
3	1, 2	anbi12d 633	. . . 4 ⊢ (𝑦 = 𝐴 → ((( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
4	3	rexbidv 3162	. . 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 7377	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 -s ( 1s /su 𝑛)) = (𝐴 -s ( 1s /su 𝑛)))
7	6	eqeq2d 2748	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
9	8	abbidv 2803	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})
10		oveq1 7377	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑛)))
11	10	eqeq2d 2748	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
12	11	rexbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
13	12	abbidv 2803	. . . . 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 7388	. . . 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 2753	. . 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 633	. 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 28503	. 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 3651	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370   No csur 27624   <s clts 27625   |s ccuts 27772   1_s c1s 27819   +_s cadds 27972   -u_s cnegs 28032   -_s csubs 28033   /_su cdivs 28200  ℕ_scnns 28326  ℝ_screno 28502

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6458  df-fv 6510  df-ov 7373  df-reno 28503

This theorem is referenced by:  reno  28505  elreno2  28508  renegscl  28511  readdscl  28512  remulscl  28515