Theorem elreno2 28440

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 28436	. 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 28439	. . . . . . . . . 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 27896	. . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
7	6	rexbidv 3158	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexab 3651	. . . . . . . . . . 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 3261	. . . . . . . . . . . 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 7389	. . . . . . . . . . . . . 14 ⊢ (𝐴 -s ( 1s /su 𝑛)) ∈ V
11		breq2 5100	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥𝑂 ≤s 𝑦 ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛))))
12	10, 11	ceqsexv 3488	. . . . . . . . . . . . 13 ⊢ (∃𝑦(𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑥𝑂 ≤s 𝑦) ↔ 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)))
13	12	rexbii 3081	. . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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		leftssno 27853	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝐴) ⊆ No
19	18	sseli 3927	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 ∈ ( L ‘𝐴) → 𝑥𝑂 ∈ No )
20	19	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝑥𝑂 ∈ No )
21	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝑥𝑂 ∈ No )
22		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
23		1sno 27798	. . . . . . . . . . . . . . . . . 18 ⊢ 1s ∈ No
24	23	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
25		nnsno 28285	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
26		nnne0s 28297	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
27	24, 25, 26	divscld 28192	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
28	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
29	22, 28	subscld 28032	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
30	21, 29, 28	sleadd1d 27965	. . . . . . . . . . . . 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))))
31		npcans 28044	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
32	22, 28, 31	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝐴)
33	32	breq2d 5108	. . . . . . . . . . . . 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))
34	30, 33	bitrd 279	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
35	34	rexbidva 3156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (∃𝑛 ∈ ℕs 𝑥𝑂 ≤s (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs (𝑥𝑂 +s ( 1s /su 𝑛)) ≤s 𝐴))
36	35	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))
37	17, 36	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))
38	37	ralbidva 3155	. . . . . . . 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))
39	5, 38	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)
40	3, 4	cofcutr2d 27897	. . . . . . . 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)
41		eqeq1 2738	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
42	41	rexbidv 3158	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))))
43	42	rexab 3651	. . . . . . . . . . . 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))
44		rexcom4 3261	. . . . . . . . . . . . 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))
45		ovex 7389	. . . . . . . . . . . . . . 15 ⊢ (𝐴 +s ( 1s /su 𝑛)) ∈ V
46		breq1 5099	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑦 ≤s 𝑥𝑂 ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂))
47	45, 46	ceqsexv 3488	. . . . . . . . . . . . . 14 ⊢ (∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
48	47	rexbii 3081	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕs ∃𝑦(𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
49		r19.41v 3164	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ ℕs (𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂))
50	49	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))
51	44, 48, 50	3bitr3ri 302	. . . . . . . . . . . 12 ⊢ (∃𝑦(∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) ∧ 𝑦 ≤s 𝑥𝑂) ↔ ∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂)
52	43, 51	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)
53		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
54		rightssno 27854	. . . . . . . . . . . . . . . . . 18 ⊢ ( R ‘𝐴) ⊆ No
55	54	sseli 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ ( R ‘𝐴) → 𝑥𝑂 ∈ No )
56	55	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝑥𝑂 ∈ No )
57	56	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → 𝑥𝑂 ∈ No )
58	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
59	57, 58	subscld 28032	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 -s ( 1s /su 𝑛)) ∈ No )
60	53, 59, 58	sleadd1d 27965	. . . . . . . . . . . . 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))))
61		npcans 28044	. . . . . . . . . . . . . . 15 ⊢ ((𝑥𝑂 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
62	57, 58, 61	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝑥𝑂 -s ( 1s /su 𝑛)) +s ( 1s /su 𝑛)) = 𝑥𝑂)
63	62	breq2d 5108	. . . . . . . . . . . . 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))
64	60, 63	bitr2d 280	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → ((𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂 ↔ 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
65	64	rexbidva 3156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (∃𝑛 ∈ ℕs (𝐴 +s ( 1s /su 𝑛)) ≤s 𝑥𝑂 ↔ ∃𝑛 ∈ ℕs 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
66	52, 65	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))))
67	66	ralbidva 3155	. . . . . . . . 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))))
68	67	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))))
69	40, 68	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)))
70	39, 69	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))))
71		lrcut 27876	. . . . . . . 8 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
72	71	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)
73		lltropt 27844	. . . . . . . . 9 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
74	73	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))
75	35	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)))
76	75, 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)
77	76	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))
78	77	ralimdva 3146	. . . . . . . . . 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))
79	78	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)
80	79	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)
81	65	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)
82	81, 52	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)
83	82	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))
84	83	ralimdva 3146	. . . . . . . . . 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))
85	84	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)
86	85	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)
87		nnsex 28279	. . . . . . . . . . . . 13 ⊢ ℕs ∈ V
88	87	abrexex 7904	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∈ V
89	88	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∈ V)
90		snexg 5378	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} ∈ V)
91		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
92	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
93	91, 92	subscld 28032	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
94		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐴 -s ( 1s /su 𝑛)) → (𝑤 ∈ No ↔ (𝐴 -s ( 1s /su 𝑛)) ∈ No ))
95	93, 94	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑤 = (𝐴 -s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
96	95	rexlimdva 3135	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
97	96	abssdv 4017	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
98		snssi 4762	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No )
99		biid 261	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No ↔ 𝐴 ∈ No )
100		vex 3442	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101	100, 7	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
102		velsn 4594	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
103		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
104	103	nnsrecgt0d 28311	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
105	104	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 0s <s ( 1s /su 𝑛))
106	92, 91	sltsubposd 28068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 0s <s ( 1s /su 𝑛) ↔ (𝐴 -s ( 1s /su 𝑛)) <s 𝐴))
107	105, 106	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) <s 𝐴)
108		breq12 5101	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → (𝑦 <s 𝑧 ↔ (𝐴 -s ( 1s /su 𝑛)) <s 𝐴))
109	107, 108	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → 𝑦 <s 𝑧))
110	109	expd 415	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑦 <s 𝑧)))
111	110	rexlimdva 3135	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑦 <s 𝑧)))
112	111	3imp 1110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = 𝐴) → 𝑦 <s 𝑧)
113	99, 101, 102, 112	syl3anb 1161	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
114	89, 90, 97, 98, 113	ssltd 27758	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝐴})
115	71	sneqd 4590	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {(( L ‘𝐴) |s ( R ‘𝐴))} = {𝐴})
116	114, 115	breqtrrd 5124	. . . . . . . . 9 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 -s ( 1s /su 𝑛))} <<s {(( L ‘𝐴) |s ( R ‘𝐴))})
117	116	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))})
118	72	sneqd 4590	. . . . . . . . 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})
119	87	abrexex 7904	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ∈ V
120	119	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ∈ V)
121	91, 92	addscld 27950	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
122		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐴 +s ( 1s /su 𝑛)) → (𝑤 ∈ No ↔ (𝐴 +s ( 1s /su 𝑛)) ∈ No ))
123	121, 122	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑤 = (𝐴 +s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
124	123	rexlimdva 3135	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛)) → 𝑤 ∈ No ))
125	124	abssdv 4017	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
126	100, 42	elab 3632	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)))
127	92, 91	sltaddpos1d 27981	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 0s <s ( 1s /su 𝑛) ↔ 𝐴 <s (𝐴 +s ( 1s /su 𝑛))))
128	105, 127	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 <s (𝐴 +s ( 1s /su 𝑛)))
129		breq12 5101	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝐴 ∧ 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → (𝑧 <s 𝑦 ↔ 𝐴 <s (𝐴 +s ( 1s /su 𝑛))))
130	128, 129	syl5ibrcom 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ((𝑧 = 𝐴 ∧ 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → 𝑧 <s 𝑦))
131	130	expcomd 416	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑧 <s 𝑦)))
132	131	rexlimdva 3135	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) → (𝑧 = 𝐴 → 𝑧 <s 𝑦)))
133	132	com23 86	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ No → (𝑧 = 𝐴 → (∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛)) → 𝑧 <s 𝑦)))
134	133	3imp 1110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑧 = 𝐴 ∧ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 +s ( 1s /su 𝑛))) → 𝑧 <s 𝑦)
135	99, 102, 126, 134	syl3anb 1161	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑧 ∈ {𝐴} ∧ 𝑦 ∈ {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑧 <s 𝑦)
136	90, 120, 98, 125, 135	ssltd 27758	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → {𝐴} <<s {𝑤 ∣ ∃𝑛 ∈ ℕs 𝑤 = (𝐴 +s ( 1s /su 𝑛))})
137	136	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))})
138	118, 137	eqbrtrd 5118	. . . . . . . 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))})
139	74, 80, 86, 117, 138	cofcut1d 27892	. . . . . . 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))}))
140	72, 139	eqtr3d 2771	. . . . . 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))}))
141	70, 140	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)))))
142		ralunb 4147	. . . . . 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))))
143		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝐴 ∈ No )
144	143, 20	subscld 28032	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝐴 -s 𝑥𝑂) ∈ No )
145		0sno 27797	. . . . . . . . . . . . . . 15 ⊢ 0s ∈ No
146	145	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ∈ No )
147		leftlt 27835	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 ∈ ( L ‘𝐴) → 𝑥𝑂 <s 𝐴)
148	147	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 𝑥𝑂 <s 𝐴)
149	20, 143	posdifsd 28067	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝑥𝑂 <s 𝐴 ↔ 0s <s (𝐴 -s 𝑥𝑂)))
150	148, 149	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s <s (𝐴 -s 𝑥𝑂))
151	146, 144, 150	sltled 27735	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → 0s ≤s (𝐴 -s 𝑥𝑂))
152		abssid 28209	. . . . . . . . . . . . 13 ⊢ (((𝐴 -s 𝑥𝑂) ∈ No ∧ 0s ≤s (𝐴 -s 𝑥𝑂)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
153	144, 151, 152	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝐴 -s 𝑥𝑂))
154	153	breq2d 5108	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s (𝐴 -s 𝑥𝑂)))
155	154	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)))
156	144	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝐴 -s 𝑥𝑂) ∈ No )
157	28, 156, 21	sleadd2d 27966	. . . . . . . . . 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))))
158		pncan3s 28042	. . . . . . . . . . . . 13 ⊢ ((𝑥𝑂 ∈ No ∧ 𝐴 ∈ No ) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
159	20, 143, 158	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
160	159	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( L ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (𝑥𝑂 +s (𝐴 -s 𝑥𝑂)) = 𝐴)
161	160	breq2d 5108	. . . . . . . . . 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))
162	155, 157, 161	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))
163	162	rexbidva 3156	. . . . . . . 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))
164	163	ralbidva 3155	. . . . . . 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))
165		absssub 28218	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ No ) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
166	55, 165	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
167	166	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝐴 -s 𝑥𝑂)) = (abss‘(𝑥𝑂 -s 𝐴)))
168		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝐴 ∈ No )
169	56, 168	subscld 28032	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (𝑥𝑂 -s 𝐴) ∈ No )
170	145	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ∈ No )
171		rightgt 27836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 ∈ ( R ‘𝐴) → 𝐴 <s 𝑥𝑂)
172	171	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 𝐴 <s 𝑥𝑂)
173	168, 56	posdifsd 28067	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (𝐴 <s 𝑥𝑂 ↔ 0s <s (𝑥𝑂 -s 𝐴)))
174	172, 173	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s <s (𝑥𝑂 -s 𝐴))
175	170, 169, 174	sltled 27735	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → 0s ≤s (𝑥𝑂 -s 𝐴))
176		abssid 28209	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝑂 -s 𝐴) ∈ No ∧ 0s ≤s (𝑥𝑂 -s 𝐴)) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
177	169, 175, 176	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
178	177	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝑥𝑂 -s 𝐴)) = (𝑥𝑂 -s 𝐴))
179	167, 178	eqtrd 2769	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (abss‘(𝐴 -s 𝑥𝑂)) = (𝑥𝑂 -s 𝐴))
180	179	breq2d 5108	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (abss‘(𝐴 -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s (𝑥𝑂 -s 𝐴)))
181	58, 57, 53	slesubd 28065	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑥𝑂 ∈ ( R ‘𝐴)) ∧ 𝑛 ∈ ℕs) → (( 1s /su 𝑛) ≤s (𝑥𝑂 -s 𝐴) ↔ 𝐴 ≤s (𝑥𝑂 -s ( 1s /su 𝑛))))
182	180, 181	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))))
183	182	rexbidva 3156	. . . . . . . 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))))
184	183	ralbidva 3155	. . . . . . 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))))
185	164, 184	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)))))
186	142, 185	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)))))
187	141, 186	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))))
188	187	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)))))
189	188	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)))))
190	1, 189	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 2712  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∪ cun 3897  {csn 4578   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356   No csur 27605   <s cslt 27606   ≤s csle 27710   <<s csslt 27747   |s cscut 27749   0_s c0s 27793   1_s c1s 27794   L cleft 27813   R cright 27814   +_s cadds 27929   -u_s cnegs 27988   -_s csubs 27989   /_su cdivs 28156  abs_scabss 28205  ℕ_scnns 28274  ℝ_screno 28434

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-dc 10354

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-ot 4587  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-nadd 8592  df-no 27608  df-slt 27609  df-bday 27610  df-sle 27711  df-sslt 27748  df-scut 27750  df-0s 27795  df-1s 27796  df-made 27815  df-old 27816  df-left 27818  df-right 27819  df-norec 27908  df-norec2 27919  df-adds 27930  df-negs 27990  df-subs 27991  df-muls 28076  df-divs 28157  df-abss 28206  df-n0s 28275  df-nns 28276  df-reno 28435

This theorem is referenced by:  0reno  28441  1reno  28442