Theorem conway 33993

Description: Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. Theorem from [Alling] p. 185. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
conway	⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem conway

Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltss1 33983	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 33981	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltss2 33984	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		ssltex2 33982	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		ssltsep 33985	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
6		noeta2 33979	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 𝑝 <s 𝑞) → ∃𝑦 ∈ No (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
7	1, 2, 3, 4, 5, 6	syl221anc 1380	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∃𝑦 ∈ No (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		3simpa 1147	. . . . . 6 ⊢ ((∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) → (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞))
9	2	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10		snex 5354	. . . . . . . . . 10 ⊢ {𝑦} ∈ V
11	9, 10	jctir 521	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
12	1	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ⊆ No )
13		snssi 4741	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → {𝑦} ⊆ No )
14	13	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → {𝑦} ⊆ No )
15	14	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17		breq2 5078	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦))
18	16, 17	ralsn 4617	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦)
19	18	ralbii 3092	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦)
20	19	biimpri 227	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
21	20	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
22	12, 15, 21	3jca 1127	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞))
23		brsslt 33980	. . . . . . . . 9 ⊢ (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
24	11, 22, 23	sylanbrc 583	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
25	24	ex 413	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → 𝐴 <<s {𝑦}))
26	4	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
27	26, 10	jctil 520	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
28	14	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
29	3	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ⊆ No )
30		ralcom 3166	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31		breq1 5077	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞))
32	16, 31	ralsn 4617	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞)
33	32	ralbii 3092	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞)
34	30, 33	sylbbr 235	. . . . . . . . . . 11 ⊢ (∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
35	34	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
36	28, 29, 35	3jca 1127	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞))
37		brsslt 33980	. . . . . . . . 9 ⊢ ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)))
38	27, 36, 37	sylanbrc 583	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
39	38	ex 413	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → (∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 → {𝑦} <<s 𝐵))
40	25, 39	anim12d 609	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → ((∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
41	8, 40	syl5 34	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → ((∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) → (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
42	41	reximdva 3203	. . . 4 ⊢ (𝐴 <<s 𝐵 → (∃𝑦 ∈ No (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) → ∃𝑦 ∈ No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
43	7, 42	mpd 15	. . 3 ⊢ (𝐴 <<s 𝐵 → ∃𝑦 ∈ No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
44		rabn0 4319	. . 3 ⊢ ({𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 ∈ No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
45	43, 44	sylibr 233	. 2 ⊢ (𝐴 <<s 𝐵 → {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46		ssrab2 4013	. . 3 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
47	46	a1i 11	. 2 ⊢ (𝐴 <<s 𝐵 → {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No )
48		simplr3 1216	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝑟 ∈ No )
49	2	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐴 ∈ V)
50		snex 5354	. . . . . . . 8 ⊢ {𝑟} ∈ V
51	49, 50	jctir 521	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → (𝐴 ∈ V ∧ {𝑟} ∈ V))
52	1	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐴 ⊆ No )
53		snssi 4741	. . . . . . . . 9 ⊢ (𝑟 ∈ No → {𝑟} ⊆ No )
54	48, 53	syl 17	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → {𝑟} ⊆ No )
55	52	sselda 3921	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
56		simplr1 1214	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝑝 ∈ No )
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑝 ∈ No )
58	48	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑟 ∈ No )
59		simplll 772	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞)) → 𝐴 <<s {𝑝})
60	59	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐴 <<s {𝑝})
61		ssltsep 33985	. . . . . . . . . . . . . 14 ⊢ (𝐴 <<s {𝑝} → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑝}𝑥 <s 𝑦)
62	60, 61	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑝}𝑥 <s 𝑦)
63	62	r19.21bi 3134	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ {𝑝}𝑥 <s 𝑦)
64		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
65		breq2 5078	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑝 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝))
66	64, 65	ralsn 4617	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝)
67	63, 66	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑝)
68		simprrl 778	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝑝 <s 𝑟)
69	68	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑝 <s 𝑟)
70	55, 57, 58, 67, 69	slttrd 33962	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → 𝑥 <s 𝑟)
71		vex 3436	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
72		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟))
73	71, 72	ralsn 4617	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟)
74	70, 73	sylibr 233	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)
75	74	ralrimiva 3103	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)
76	52, 54, 75	3jca 1127	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → (𝐴 ⊆ No ∧ {𝑟} ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦))
77		brsslt 33980	. . . . . . 7 ⊢ (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑟} ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
78	51, 76, 77	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐴 <<s {𝑟})
79	4	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐵 ∈ V)
80	79, 50	jctil 520	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ({𝑟} ∈ V ∧ 𝐵 ∈ V))
81	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝐵 ⊆ No )
82	48	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑟 ∈ No )
83		simplr2 1215	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝑞 ∈ No )
84	83	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑞 ∈ No )
85	81	sselda 3921	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ No )
86		simprrr 779	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → 𝑟 <s 𝑞)
87	86	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑟 <s 𝑞)
88		simplrr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞)) → {𝑞} <<s 𝐵)
89	88	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → {𝑞} <<s 𝐵)
90		ssltsep 33985	. . . . . . . . . . . . . 14 ⊢ ({𝑞} <<s 𝐵 → ∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
92		vex 3436	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
93		breq1 5077	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (𝑥 <s 𝑦 ↔ 𝑞 <s 𝑦))
94	93	ralbidv 3112	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦))
95	92, 94	ralsn 4617	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦)
96	91, 95	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦)
97	96	r19.21bi 3134	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑞 <s 𝑦)
98	82, 84, 85, 87, 97	slttrd 33962	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) ∧ 𝑦 ∈ 𝐵) → 𝑟 <s 𝑦)
99	98	ralrimiva 3103	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦)
100		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑟 <s 𝑦))
101	100	ralbidv 3112	. . . . . . . . . 10 ⊢ (𝑥 = 𝑟 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦))
102	71, 101	ralsn 4617	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦)
103	99, 102	sylibr 233	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
104	54, 81, 103	3jca 1127	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → ({𝑟} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
105		brsslt 33980	. . . . . . 7 ⊢ ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
106	80, 104, 105	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → {𝑟} <<s 𝐵)
107	48, 78, 106	jca32 516	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞))) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
108	107	exp44 438	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑝 ∈ No ∧ 𝑞 ∈ No ∧ 𝑟 ∈ No )) → ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
109	108	ralrimivvva 3127	. . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
110		sneq 4571	. . . . . . 7 ⊢ (𝑦 = 𝑝 → {𝑦} = {𝑝})
111	110	breq2d 5086	. . . . . 6 ⊢ (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
112	110	breq1d 5084	. . . . . 6 ⊢ (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
113	111, 112	anbi12d 631	. . . . 5 ⊢ (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
114	113	ralrab 3630	. . . 4 ⊢ (∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))) ↔ ∀𝑝 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
115		sneq 4571	. . . . . . . . 9 ⊢ (𝑦 = 𝑞 → {𝑦} = {𝑞})
116	115	breq2d 5086	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
117	115	breq1d 5084	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
118	116, 117	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
119	118	ralrab 3630	. . . . . 6 ⊢ (∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))) ↔ ∀𝑞 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
120		sneq 4571	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑟 → {𝑦} = {𝑟})
121	120	breq2d 5086	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
122	120	breq1d 5084	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
123	121, 122	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124	123	elrab 3624	. . . . . . . . 9 ⊢ (𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
125	124	imbi2i 336	. . . . . . . 8 ⊢ (((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
126	125	ralbii 3092	. . . . . . 7 ⊢ (∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
127	126	ralbii 3092	. . . . . 6 ⊢ (∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
128		r19.21v 3113	. . . . . . 7 ⊢ (∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))) ↔ ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
129	128	ralbii 3092	. . . . . 6 ⊢ (∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))) ↔ ∀𝑞 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
130	119, 127, 129	3bitr4i 303	. . . . 5 ⊢ (∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
131	130	ralbii 3092	. . . 4 ⊢ (∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
132		r19.21v 3113	. . . . . . 7 ⊢ (∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))) ↔ ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
133	132	ralbii 3092	. . . . . 6 ⊢ (∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))) ↔ ∀𝑞 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
134		r19.21v 3113	. . . . . 6 ⊢ (∀𝑞 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))) ↔ ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
135	133, 134	bitri 274	. . . . 5 ⊢ (∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))) ↔ ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
136	135	ralbii 3092	. . . 4 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))) ↔ ∀𝑝 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
137	114, 131, 136	3bitr4i 303	. . 3 ⊢ (∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
138	109, 137	sylibr 233	. 2 ⊢ (𝐴 <<s 𝐵 → ∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
139		nocvxmin 33973	. 2 ⊢ (({𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ∧ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No ∧ ∀𝑝 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 ∈ No ((𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
140	45, 47, 138, 139	syl3anc 1370	1 ⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  {crab 3068  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ cuni 4839  ∩ cint 4879   class class class wbr 5074   “ cima 5592  suc csuc 6268  ‘cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845   <<s csslt 33975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976

This theorem is referenced by:  scutcut  33995  scutbday  33998  eqscut  33999  scutun12  34004  scutbdaylt  34012