Theorem conway 33259

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. (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 33252	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		ssltex1 33250	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		ssltss2 33253	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		ssltex2 33251	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		ssltsep 33254	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
6		noeta 33217	. . . . 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 1377	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∃𝑦 ∈ No (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		3simpa 1144	. . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10		snex 5323	. . . . . . . . . 10 ⊢ {𝑦} ∈ V
11	9, 10	jctir 523	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
12	1	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ⊆ No )
13		snssi 4734	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → {𝑦} ⊆ No )
14	13	adantl 484	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → {𝑦} ⊆ No )
15	14	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16		vex 3497	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17		breq2 5062	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦))
18	16, 17	ralsn 4612	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦)
19	18	ralbii 3165	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦)
20	19	biimpri 230	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
21	20	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
22	12, 15, 21	3jca 1124	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞))
23		brsslt 33249	. . . . . . . . 9 ⊢ (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
24	11, 22, 23	sylanbrc 585	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
25	24	ex 415	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → 𝐴 <<s {𝑦}))
26	4	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
27	26, 10	jctil 522	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
28	14	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
29	3	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ⊆ No )
30		ralcom 3354	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31		breq1 5061	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞))
32	16, 31	ralsn 4612	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞)
33	32	ralbii 3165	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞)
34	30, 33	sylbbr 238	. . . . . . . . . . 11 ⊢ (∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
35	34	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
36	28, 29, 35	3jca 1124	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞))
37		brsslt 33249	. . . . . . . . 9 ⊢ ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)))
38	27, 36, 37	sylanbrc 585	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
39	38	ex 415	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → (∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 → {𝑦} <<s 𝐵))
40	25, 39	anim12d 610	. . . . . 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 3274	. . . 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 4338	. . 3 ⊢ ({𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 ∈ No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
45	43, 44	sylibr 236	. 2 ⊢ (𝐴 <<s 𝐵 → {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46		ssrab2 4055	. . 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 1213	. . . . . 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 724	. . . . . . . 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 5323	. . . . . . . 8 ⊢ {𝑟} ∈ V
51	49, 50	jctir 523	. . . . . . 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 724	. . . . . . . 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 4734	. . . . . . . . 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 3966	. . . . . . . . . . 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 1211	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 483	. . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 33254	. . . . . . . . . . . . . 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 3208	. . . . . . . . . . . 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 3497	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
65		breq2 5062	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑝 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝))
66	64, 65	ralsn 4612	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝)
67	63, 66	sylib 220	. . . . . . . . . . 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 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))) → 𝑝 <s 𝑟)
69	68	adantr 483	. . . . . . . . . . 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 33233	. . . . . . . . . 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 3497	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
72		breq2 5062	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟))
73	71, 72	ralsn 4612	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟)
74	70, 73	sylibr 236	. . . . . . . . 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 3182	. . . . . . . 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 1124	. . . . . . 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 33249	. . . . . . 7 ⊢ (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑟} ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
78	51, 76, 77	sylanbrc 585	. . . . . 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 724	. . . . . . . 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 522	. . . . . . 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 724	. . . . . . . 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 483	. . . . . . . . . . 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 1212	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 3966	. . . . . . . . . . 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 780	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 33254	. . . . . . . . . . . . . 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 3497	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
93		breq1 5061	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (𝑥 <s 𝑦 ↔ 𝑞 <s 𝑦))
94	93	ralbidv 3197	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦))
95	92, 94	ralsn 4612	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦)
96	91, 95	sylib 220	. . . . . . . . . . . 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 3208	. . . . . . . . . . 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 33233	. . . . . . . . . 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 3182	. . . . . . . . 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 5061	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑟 <s 𝑦))
101	100	ralbidv 3197	. . . . . . . . . 10 ⊢ (𝑥 = 𝑟 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦))
102	71, 101	ralsn 4612	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦)
103	99, 102	sylibr 236	. . . . . . . 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 1124	. . . . . . 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 33249	. . . . . . 7 ⊢ ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
106	80, 104, 105	sylanbrc 585	. . . . . 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 518	. . . . 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 440	. . . 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 3192	. . 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 4570	. . . . . . 7 ⊢ (𝑦 = 𝑝 → {𝑦} = {𝑝})
111	110	breq2d 5070	. . . . . 6 ⊢ (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
112	110	breq1d 5068	. . . . . 6 ⊢ (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
113	111, 112	anbi12d 632	. . . . 5 ⊢ (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
114	113	ralrab 3684	. . . 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 4570	. . . . . . . . 9 ⊢ (𝑦 = 𝑞 → {𝑦} = {𝑞})
116	115	breq2d 5070	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
117	115	breq1d 5068	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
118	116, 117	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
119	118	ralrab 3684	. . . . . 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 4570	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑟 → {𝑦} = {𝑟})
121	120	breq2d 5070	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
122	120	breq1d 5068	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
123	121, 122	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124	123	elrab 3679	. . . . . . . . 9 ⊢ (𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
125	124	imbi2i 338	. . . . . . . 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 3165	. . . . . . 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 3165	. . . . . 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 3175	. . . . . . 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 3165	. . . . . 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 305	. . . . 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 3165	. . . 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 3175	. . . . . . 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 3165	. . . . . 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 3175	. . . . . 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 277	. . . . 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 3165	. . . 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 305	. . 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 236	. 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 33243	. 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 1367	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 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  {csn 4560  ∪ cuni 4831  ∩ cint 4868   class class class wbr 5058   “ cima 5552  suc csuc 6187  ‘cfv 6349   No csur 33142   <s cslt 33143   bday cbday 33144   <<s csslt 33245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-1o 8096  df-2o 8097  df-no 33145  df-slt 33146  df-bday 33147  df-sslt 33246

This theorem is referenced by:  scutcut  33261  scutbday  33262  scutun12  33266  scutbdaylt  33271