Theorem elreno2 28491

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 28487	. 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 28490	. . . . . . . . . 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 27921	. . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
7	6	rexbidv 3160	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexab 3653	. . . . . . . . . . 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 3263	. . . . . . . . . . . 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 7391	. . . . . . . . . . . . . 14 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
11		breq2 5102	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥𝑂 ≤s 𝑦 ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛))))
12	10, 11	ceqsexv 3490	. . . . . . . . . . . . 13 ⊢ (∃𝑦(𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
13	12	rexbii 3083	. . . . . . . . . . . 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 3166	. . . . . . . . . . . . 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 1849	. . . . . . . . . . . 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 27873	. . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
22		1no 27806	. . . . . . . . . . . . . . . . . 18 ⊢ 1s ∈ No
23	22	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
24		nnno 28320	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
25		nnne0s 28333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
26	23, 24, 25	divscld 28220	. . . . . . . . . . . . . . . 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 28059	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
29	20, 28, 27	leadds1d 27991	. . . . . . . . . . . . 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 28071	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
31	21, 27, 30	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
32	31	breq2d 5110	. . . . . . . . . . . . 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 3158	. . . . . . . . . . 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 715	. . . . . . . . . 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 3157	. . . . . . . 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 27922	. . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
41	40	rexbidv 3160	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
42	41	rexab 3653	. . . . . . . . . . . 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 3263	. . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
45		breq1 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑦 ≤s 𝑥𝑂 ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂))
46	44, 45	ceqsexv 3490	. . . . . . . . . . . . . 14 ⊢ (∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
47	46	rexbii 3083	. . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . 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 1849	. . . . . . . . . . . . 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 766	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
53		rightno 27874	. . . . . . . . . . . . . . . . 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 28059	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 -s ( 1s /su 𝑛)) ∈ No )
58	52, 57, 56	leadds1d 27991	. . . . . . . . . . . . 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 28071	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝑂 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
60	55, 56, 59	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
61	60	breq2d 5110	. . . . . . . . . . . . 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 3158	. . . . . . . . . . 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 3157	. . . . . . . . 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 27900	. . . . . . . 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 27858	. . . . . . . . 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 3148	. . . . . . . . . 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 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 ∈ ( 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 3148	. . . . . . . . . 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 716	. . . . . . . 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 28314	. . . . . . . . . . . . 13 ⊢ ℕs ∈ V
86	85	abrexex 7906	. . . . . . . . . . . 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 5380	. . . . . . . . . . 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 28059	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
92		eleq1 2824	. . . . . . . . . . . . . 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 3137	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
95	94	abssdv 4019	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
96		snssi 4764	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No )
97		biid 261	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No ↔ 𝐴 ∈ No )
98		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
99	98, 7	elab 3634	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
100		velsn 4596	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
101		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
102	101	nnsrecgt0d 28347	. . . . . . . . . . . . . . . . . 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 28095	. . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑦 <s 𝑧)))
110	109	3imp 1110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → 𝑦 <s 𝑧)
111	97, 99, 100, 110	syl3anb 1161	. . . . . . . . . . 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 27764	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝐴})
113	69	sneqd 4592	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {(( L ‘𝐴) |s ( R ‘𝐴))} = {𝐴})
114	112, 113	breqtrrd 5126	. . . . . . . . 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 4592	. . . . . . . . 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 7906	. . . . . . . . . . . 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 27976	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
120		eleq1 2824	. . . . . . . . . . . . . 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 3137	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
123	122	abssdv 4019	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
124	98, 41	elab 3634	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)))
125	90, 89	ltaddspos1d 28007	. . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . . 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 1110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑧 = 𝐴 ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → 𝑧 <s 𝑦)
133	97, 100, 124, 132	syl3anb 1161	. . . . . . . . . . 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 27764	. . . . . . . . . 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 5120	. . . . . . . 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 27917	. . . . . . 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 2773	. . . . . 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 800	. . . . 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 4149	. . . . . 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 28059	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝐴 -s 𝑥𝑂) ∈ No )
143		0no 27805	. . . . . . . . . . . . . . 15 ⊢ 0s ∈ No
144	143	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ∈ No )
145		leftlt 27849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 ∈ ( L ‘𝐴) → 𝑥𝑂 <s 𝐴)
146	145	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝑥𝑂 <s 𝐴)
147	19, 141	posdifsd 28094	. . . . . . . . . . . . . . 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 27741	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ≤s (𝐴 -s 𝑥𝑂))
150		abssid 28237	. . . . . . . . . . . . 13 ⊢ (((𝐴 -s 𝑥𝑂) ∈ No ∧ 0s ≤s (𝐴 -s 𝑥𝑂)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
151	142, 149, 150	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
152	151	breq2d 5110	. . . . . . . . . . 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 27992	. . . . . . . . . 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 28069	. . . . . . . . . . . . 13 ⊢ ((𝑥𝑂 ∈ No ∧ 𝐴 ∈ No ) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
157	19, 141, 156	syl2anc 584	. . . . . . . . . . . 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 5110	. . . . . . . . . 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 3158	. . . . . . . 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 3157	. . . . . . 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 28246	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ No ) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
164	53, 163	sylan2 593	. . . . . . . . . . . . 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 28059	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (𝑥𝑂 -s 𝐴) ∈ No )
168	143	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ∈ No )
169		rightgt 27850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑂)
170	169	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑥𝑂)
171	166, 54	posdifsd 28094	. . . . . . . . . . . . . . . 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 27741	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ≤s (𝑥𝑂 -s 𝐴))
174		abssid 28237	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑂 -s 𝐴) ∈ No ∧ 0s ≤s (𝑥𝑂 -s 𝐴)) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
175	167, 173, 174	syl2anc 584	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝑥𝑂 -s 𝐴))
178	177	breq2d 5110	. . . . . . . . . 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 28092	. . . . . . . . . 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 3158	. . . . . . . 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 3157	. . . . . . 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 632	. . . . . 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 630	. . 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 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∪ cun 3899  {csn 4580   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358   No csur 27607   <s clts 27608   ≤s cles 27712   <<s cslts 27753   |s ccuts 27755   0_s c0s 27801   1_s c1s 27802   L cleft 27821   R cright 27822   +_s cadds 27955   -u_s cnegs 28015   -_s csubs 28016   /_su cdivs 28183  abs_scabss 28233  ℕ_scnns 28309  ℝ_screno 28485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-dc 10356

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103  df-divs 28184  df-abss 28234  df-n0s 28310  df-nns 28311  df-reno 28486

This theorem is referenced by:  0reno  28492  1reno  28493