Theorem elreno2 28503

Description: Alternate characterization of the surreal reals. Theorem 4.4(b) of [Gonshor] p. 39. (Contributed by Scott Fenton, 29-Jan-2026.)

Assertion

Ref	Expression
elreno2	⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)))))

Distinct variable group:   𝐴,𝑛,𝑥_𝑂

Proof of Theorem elreno2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreno 28499	. 2 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}))))
2		recut 28502	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
3	2	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
4		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}))
5	3, 4	cofcutr1d 27933	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦)
6		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
7	6	rexbidv 3162	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexab 3655	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦 ↔ ∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦))
9		rexcom4 3265	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs ∃𝑦(𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ ∃𝑦∃𝑛 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦))
10		ovex 7401	. . . . . . . . . . . . . 14 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
11		breq2 5104	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥𝑂 ≤s 𝑦 ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛))))
12	10, 11	ceqsexv 3492	. . . . . . . . . . . . 13 ⊢ (∃𝑦(𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
13	12	rexbii 3085	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕs ∃𝑦(𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ ∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
14		r19.41v 3168	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦))
15	14	exbii 1850	. . . . . . . . . . . 12 ⊢ (∃𝑦∃𝑛 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ ∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦))
16	9, 13, 15	3bitr3ri 302	. . . . . . . . . . 11 ⊢ (∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ ∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
17	8, 16	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
18		leftno 27885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 ∈ ( L ‘𝐴) → 𝑥𝑂 ∈ No )
19	18	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝑥𝑂 ∈ No )
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝑥𝑂 ∈ No )
21		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
22		1no 27818	. . . . . . . . . . . . . . . . . 18 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
24		nnno 28332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
25		nnne0s 28345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
26	23, 24, 25	divscld 28232	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
27	26	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
28	21, 27	subscld 28071	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
29	20, 28, 27	leadds1d 28003	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛))))
30		npcans 28083	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
31	21, 27, 30	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
32	31	breq2d 5112	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝑥𝑂 +s ( 1s /su 𝑛)) ≤s ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
33	29, 32	bitrd 279	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
34	33	rexbidva 3160	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
35	34	adantlr 716	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
36	17, 35	bitrid 283	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦 ↔ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
37	36	ralbidva 3159	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦 ↔ ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
38	5, 37	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴)
39	3, 4	cofcutr2d 27934	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂)
40		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
41	40	rexbidv 3162	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
42	41	rexab 3655	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂 ↔ ∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂))
43		rexcom4 3265	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑦∃𝑛 ∈ ℕs (𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂))
44		ovex 7401	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
45		breq1 5103	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑦 ≤s 𝑥𝑂 ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂))
46	44, 45	ceqsexv 3492	. . . . . . . . . . . . . 14 ⊢ (∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
47	46	rexbii 3085	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
48		r19.41v 3168	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂))
49	48	exbii 1850	. . . . . . . . . . . . 13 ⊢ (∃𝑦∃𝑛 ∈ ℕs (𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂))
50	43, 47, 49	3bitr3ri 302	. . . . . . . . . . . 12 ⊢ (∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
51	42, 50	bitri 275	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂 ↔ ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
52		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
53		rightno 27886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ ( R ‘𝐴) → 𝑥𝑂 ∈ No )
54	53	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝑥𝑂 ∈ No )
55	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝑥𝑂 ∈ No )
56	26	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
57	55, 56	subscld 28071	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 -s ( 1s /su 𝑛)) ∈ No )
58	52, 57, 56	leadds1d 28003	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)) ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛))))
59		npcans 28083	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝑂 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
60	55, 56, 59	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
61	60	breq2d 5112	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) ≤s ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂))
62	58, 61	bitr2d 280	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂 ↔ 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
63	62	rexbidva 3160	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂 ↔ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
64	51, 63	bitrid 283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂 ↔ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
65	64	ralbidva 3159	. . . . . . . . 9 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂 ↔ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
66	65	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → (∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂 ↔ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
67	39, 66	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))
68	38, 67	jca 511	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) → (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
69		lrcut 27912	. . . . . . . 8 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
70	69	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
71		lltr 27870	. . . . . . . . 9 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
72	71	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → ( L ‘𝐴) <<s ( R ‘𝐴))
73	34	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴) → ∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
74	73, 17	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴) → ∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦)
75	74	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 → ∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦))
76	75	ralimdva 3150	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 → ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦))
77	76	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴) → ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦)
78	77	adantrr 718	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))}𝑥𝑂 ≤s 𝑦)
79	63	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))) → ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
80	79, 51	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))) → ∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂)
81	80	ex 412	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)) → ∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂))
82	81	ralimdva 3150	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)) → ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂))
83	82	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))) → ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂)
84	83	adantrl 717	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}𝑦 ≤s 𝑥𝑂)
85		nnsex 28326	. . . . . . . . . . . . 13 ⊢ ℕs ∈ V
86	85	abrexex 7916	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∈ V
87	86	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∈ V)
88		snexg 5386	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} ∈ V)
89		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
90	26	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
91	89, 90	subscld 28071	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
92		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐴 -s ( 1s /su 𝑛)) → (𝑤 ∈ No ↔ (𝐴 -s ( 1s /su 𝑛)) ∈ No ))
93	91, 92	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑤 = (𝐴 -s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
94	93	rexlimdva 3139	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
95	94	abssdv 4021	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
96		snssi 4766	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No )
97		biid 261	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No ↔ 𝐴 ∈ No )
98		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
99	98, 7	elab 3636	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
100		velsn 4598	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
101		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
102	101	nnsrecgt0d 28359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
103	102	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 0s <s ( 1s /su 𝑛))
104	90, 89	ltsubsposd 28107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 0s <s ( 1s /su 𝑛) ↔ (𝐴 -s ( 1s /su 𝑛)) <s 𝐴))
105	103, 104	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) <s 𝐴)
106		breq12 5105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → (𝑦 <s 𝑧 ↔ (𝐴 -s ( 1s /su 𝑛)) <s 𝐴))
107	105, 106	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → 𝑦 <s 𝑧))
108	107	expd 415	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑦 <s 𝑧)))
109	108	rexlimdva 3139	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑦 <s 𝑧)))
110	109	3imp 1111	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → 𝑦 <s 𝑧)
111	97, 99, 100, 110	syl3anb 1162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
112	87, 88, 95, 96, 111	sltsd 27776	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝐴})
113	69	sneqd 4594	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {(( L ‘𝐴) |s ( R ‘𝐴))} = {𝐴})
114	112, 113	breqtrrd 5128	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {(( L ‘𝐴) |s ( R ‘𝐴))})
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {(( L ‘𝐴) |s ( R ‘𝐴))})
116	70	sneqd 4594	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → {(( L ‘𝐴) |s ( R ‘𝐴))} = {𝐴})
117	85	abrexex 7916	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ∈ V
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ∈ V)
119	89, 90	addscld 27988	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
120		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐴 +s ( 1s /su 𝑛)) → (𝑤 ∈ No ↔ (𝐴 +s ( 1s /su 𝑛)) ∈ No ))
121	119, 120	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑤 = (𝐴 +s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
122	121	rexlimdva 3139	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
123	122	abssdv 4021	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
124	98, 41	elab 3636	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)))
125	90, 89	ltaddspos1d 28019	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 0s <s ( 1s /su 𝑛) ↔ 𝐴 <s (𝐴 +s ( 1s /su 𝑛))))
126	103, 125	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 <s (𝐴 +s ( 1s /su 𝑛)))
127		breq12 5105	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝐴 ∧ 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → (𝑧 <s 𝑦 ↔ 𝐴 <s (𝐴 +s ( 1s /su 𝑛))))
128	126, 127	syl5ibrcom 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((𝑧 = 𝐴 ∧ 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → 𝑧 <s 𝑦))
129	128	expcomd 416	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑧 <s 𝑦)))
130	129	rexlimdva 3139	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑧 <s 𝑦)))
131	130	com23 86	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (𝑧 = 𝐴 → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) → 𝑧 <s 𝑦)))
132	131	3imp 1111	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑧 = 𝐴 ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → 𝑧 <s 𝑦)
133	97, 100, 124, 132	syl3anb 1162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑧 ∈ {𝐴} ∧ 𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑧 <s 𝑦)
134	88, 118, 96, 123, 133	sltsd 27776	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝐴} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
135	134	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → {𝐴} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
136	116, 135	eqbrtrd 5122	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → {(( L ‘𝐴) |s ( R ‘𝐴))} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
137	72, 78, 84, 115, 136	cofcut1d 27929	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → (( L ‘𝐴) |s ( R ‘𝐴)) = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}))
138	70, 137	eqtr3d 2774	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))) → 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}))
139	68, 138	impbida 801	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}) ↔ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))))
140		ralunb 4151	. . . . . 6 ⊢ (∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂))))
141		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
142	141, 19	subscld 28071	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝐴 -s 𝑥𝑂) ∈ No )
143		0no 27817	. . . . . . . . . . . . . . 15 ⊢ 0s ∈ No
144	143	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ∈ No )
145		leftlt 27861	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 ∈ ( L ‘𝐴) → 𝑥𝑂 <s 𝐴)
146	145	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝑥𝑂 <s 𝐴)
147	19, 141	posdifsd 28106	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝑥𝑂 <s 𝐴 ↔ 0s <s (𝐴 -s 𝑥𝑂)))
148	146, 147	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s <s (𝐴 -s 𝑥𝑂))
149	144, 142, 148	ltlesd 27753	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ≤s (𝐴 -s 𝑥𝑂))
150		abssid 28249	. . . . . . . . . . . . 13 ⊢ (((𝐴 -s 𝑥𝑂) ∈ No ∧ 0s ≤s (𝐴 -s 𝑥𝑂)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
151	142, 149, 150	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
152	151	breq2d 5112	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s (𝐴 -s 𝑥𝑂)))
153	152	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s (𝐴 -s 𝑥𝑂)))
154	142	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 -s 𝑥𝑂) ∈ No )
155	27, 154, 20	leadds2d 28004	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (𝐴 -s 𝑥𝑂) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s (𝑥𝑂 +s (𝐴 -s 𝑥𝑂))))
156		pncan3s 28081	. . . . . . . . . . . . 13 ⊢ ((𝑥𝑂 ∈ No ∧ 𝐴 ∈ No ) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
157	19, 141, 156	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
158	157	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
159	158	breq2d 5112	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝑥𝑂 +s ( 1s /su 𝑛)) ≤s (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
160	153, 155, 159	3bitrd 305	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
161	160	rexbidva 3160	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
162	161	ralbidva 3159	. . . . . . 7 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
163		abssubs 28258	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ No ) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
164	53, 163	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
166		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
167	54, 166	subscld 28071	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (𝑥𝑂 -s 𝐴) ∈ No )
168	143	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ∈ No )
169		rightgt 27862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑂)
170	169	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑥𝑂)
171	166, 54	posdifsd 28106	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑥𝑂 ↔ 0s <s (𝑥𝑂 -s 𝐴)))
172	170, 171	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s <s (𝑥𝑂 -s 𝐴))
173	168, 167, 172	ltlesd 27753	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ≤s (𝑥𝑂 -s 𝐴))
174		abssid 28249	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑂 -s 𝐴) ∈ No ∧ 0s ≤s (𝑥𝑂 -s 𝐴)) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
175	167, 173, 174	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
176	175	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
177	165, 176	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝑥𝑂 -s 𝐴))
178	177	breq2d 5112	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s (𝑥𝑂 -s 𝐴)))
179	56, 55, 52	lesubsd 28104	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (𝑥𝑂 -s 𝐴) ↔ 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
180	178, 179	bitrd 279	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
181	180	rexbidva 3160	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
182	181	ralbidva 3159	. . . . . . 7 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
183	162, 182	anbi12d 633	. . . . . 6 ⊢ (𝐴 ∈ No → ((∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂))) ↔ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))))
184	140, 183	bitrid 283	. . . . 5 ⊢ (𝐴 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ (∀𝑥𝑂 ∈ ( L ‘𝐴)∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴 ∧ ∀𝑥𝑂 ∈ ( R ‘𝐴)∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛)))))
185	139, 184	bitr4d 282	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂))))
186	185	anbi2d 631	. . 3 ⊢ (𝐴 ∈ No → ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})) ↔ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)))))
187	186	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}))) ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)))))
188	1, 187	bitri 275	1 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∪ cun 3901  {csn 4582   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368   No csur 27619   <s clts 27620   ≤s cles 27724   <<s cslts 27765   |s ccuts 27767   0_s c0s 27813   1_s c1s 27814   L cleft 27833   R cright 27834   +_s cadds 27967   -u_s cnegs 28027   -_s csubs 28028   /_su cdivs 28195  abs_scabss 28245  ℕ_scnns 28321  ℝ_screno 28497

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-dc 10368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-divs 28196  df-abss 28246  df-n0s 28322  df-nns 28323  df-reno 28498

This theorem is referenced by:  0reno  28504  1reno  28505