Theorem conway 27771

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		sltsss1 27757	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
2		sltsex1 27755	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
3		sltsss2 27758	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
4		sltsex2 27756	. . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
5		sltssep 27759	. . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
6		noeta2 27753	. . . . 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 1384	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∃𝑦 ∈ No (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 ∧ ( bday ‘𝑦) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
8		3simpa 1149	. . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10		vsnex 5378	. . . . . . . . . 10 ⊢ {𝑦} ∈ V
11	9, 10	jctir 520	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
12	1	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 ⊆ No )
13		snssi 4730	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → {𝑦} ⊆ No )
14	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → {𝑦} ⊆ No )
15	14	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17		breq2 5090	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦))
18	16, 17	ralsn 4626	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑝 <s 𝑦)
19	18	ralbii 3084	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦)
20	19	biimpri 228	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
21	20	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)
22	12, 15, 21	3jca 1129	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞))
23		brslts 27754	. . . . . . . . 9 ⊢ (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑦} ⊆ No ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
24	11, 22, 23	sylanbrc 584	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑝 ∈ 𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
25	24	ex 412	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) → (∀𝑝 ∈ 𝐴 𝑝 <s 𝑦 → 𝐴 <<s {𝑦}))
26	4	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
27	26, 10	jctil 519	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
28	14	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
29	3	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → 𝐵 ⊆ No )
30		ralcom 3266	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31		breq1 5089	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑦 → (𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞))
32	16, 31	ralsn 4626	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ 𝑦 <s 𝑞)
33	32	ralbii 3084	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐵 ∀𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞)
34	30, 33	sylbbr 236	. . . . . . . . . . 11 ⊢ (∀𝑞 ∈ 𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
35	34	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)
36	28, 29, 35	3jca 1129	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞))
37		brslts 27754	. . . . . . . . 9 ⊢ ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑝 ∈ {𝑦}∀𝑞 ∈ 𝐵 𝑝 <s 𝑞)))
38	27, 36, 37	sylanbrc 584	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑦 ∈ No ) ∧ ∀𝑞 ∈ 𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
39	38	ex 412	. . . . . . 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 3151	. . . 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 4330	. . 3 ⊢ ({𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 ∈ No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
45	43, 44	sylibr 234	. 2 ⊢ (𝐴 <<s 𝐵 → {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46		ssrab2 4021	. . 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 1219	. . . . . 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 727	. . . . . . . 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		vsnex 5378	. . . . . . . 8 ⊢ {𝑟} ∈ V
51	49, 50	jctir 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))) → (𝐴 ∈ V ∧ {𝑟} ∈ V))
52	1	ad2antrr 727	. . . . . . . 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	48	snssd 4731	. . . . . . . 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 )
54	52	sselda 3922	. . . . . . . . . . 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 )
55		simplr1 1217	. . . . . . . . . . . 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 )
56	55	adantr 480	. . . . . . . . . . 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 )
57	48	adantr 480	. . . . . . . . . . 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 )
58		simplll 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞)) → 𝐴 <<s {𝑝})
59	58	adantl 481	. . . . . . . . . . . . . 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})
60		sltssep 27759	. . . . . . . . . . . . . 14 ⊢ (𝐴 <<s {𝑝} → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑝}𝑥 <s 𝑦)
61	59, 60	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)
62	61	r19.21bi 3230	. . . . . . . . . . . 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)
63		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
64		breq2 5090	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑝 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝))
65	63, 64	ralsn 4626	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑝)
66	62, 65	sylib 218	. . . . . . . . . . 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)
67		simprrl 781	. . . . . . . . . . . 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)
68	67	adantr 480	. . . . . . . . . . 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)
69	54, 56, 57, 66, 68	ltstrd 27727	. . . . . . . . . 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)
70		vex 3434	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
71		breq2 5090	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟))
72	70, 71	ralsn 4626	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑟)
73	69, 72	sylibr 234	. . . . . . . . 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)
74	73	ralrimiva 3130	. . . . . . . 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)
75	52, 53, 74	3jca 1129	. . . . . . 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))
76		brslts 27754	. . . . . . 7 ⊢ (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 ⊆ No ∧ {𝑟} ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
77	51, 75, 76	sylanbrc 584	. . . . . 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})
78	4	ad2antrr 727	. . . . . . . 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)
79	78, 50	jctil 519	. . . . . . 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))
80	3	ad2antrr 727	. . . . . . . 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 )
81	48	adantr 480	. . . . . . . . . . 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 )
82		simplr2 1218	. . . . . . . . . . . 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 )
83	82	adantr 480	. . . . . . . . . . 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 )
84	80	sselda 3922	. . . . . . . . . . 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 )
85		simprrr 782	. . . . . . . . . . . 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)
86	85	adantr 480	. . . . . . . . . . 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)
87		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟 ∧ 𝑟 <s 𝑞)) → {𝑞} <<s 𝐵)
88	87	adantl 481	. . . . . . . . . . . . . 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)
89		sltssep 27759	. . . . . . . . . . . . . 14 ⊢ ({𝑞} <<s 𝐵 → ∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)
90	88, 89	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)
91		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
92		breq1 5089	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑞 → (𝑥 <s 𝑦 ↔ 𝑞 <s 𝑦))
93	92	ralbidv 3161	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦))
94	91, 93	ralsn 4626	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑞}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑞 <s 𝑦)
95	90, 94	sylib 218	. . . . . . . . . . . 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)
96	95	r19.21bi 3230	. . . . . . . . . . 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)
97	81, 83, 84, 86, 96	ltstrd 27727	. . . . . . . . . 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)
98	97	ralrimiva 3130	. . . . . . . . 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)
99		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → (𝑥 <s 𝑦 ↔ 𝑟 <s 𝑦))
100	99	ralbidv 3161	. . . . . . . . . 10 ⊢ (𝑥 = 𝑟 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦))
101	70, 100	ralsn 4626	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑟 <s 𝑦)
102	98, 101	sylibr 234	. . . . . . . 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)
103	53, 80, 102	3jca 1129	. . . . . . 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))
104		brslts 27754	. . . . . . 7 ⊢ ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ {𝑟}∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
105	79, 103, 104	sylanbrc 584	. . . . . 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)
106	48, 77, 105	jca32 515	. . . . 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)))
107	106	exp44 437	. . . 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))))))
108	107	ralrimivvva 3184	. . 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))))))
109		sneq 4578	. . . . . . 7 ⊢ (𝑦 = 𝑝 → {𝑦} = {𝑝})
110	109	breq2d 5098	. . . . . 6 ⊢ (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
111	109	breq1d 5096	. . . . . 6 ⊢ (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
112	110, 111	anbi12d 633	. . . . 5 ⊢ (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
113	112	ralrab 3641	. . . 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))))))
114		sneq 4578	. . . . . . . . 9 ⊢ (𝑦 = 𝑞 → {𝑦} = {𝑞})
115	114	breq2d 5098	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
116	114	breq1d 5096	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
117	115, 116	anbi12d 633	. . . . . . 7 ⊢ (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
118	117	ralrab 3641	. . . . . 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)))))
119		sneq 4578	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑟 → {𝑦} = {𝑟})
120	119	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
121	119	breq1d 5096	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
122	120, 121	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
123	122	elrab 3635	. . . . . . . . 9 ⊢ (𝑟 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 ∈ No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124	123	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))))
125	124	ralbii 3084	. . . . . . 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))))
126	125	ralbii 3084	. . . . . 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))))
127		r19.21v 3163	. . . . . . 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)))))
128	127	ralbii 3084	. . . . . 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)))))
129	118, 126, 128	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)))))
130	129	ralbii 3084	. . . 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)))))
131		r19.21v 3163	. . . . . . 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))))))
132	131	ralbii 3084	. . . . . 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))))))
133		r19.21v 3163	. . . . . 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))))))
134	132, 133	bitri 275	. . . . 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))))))
135	134	ralbii 3084	. . . 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))))))
136	113, 130, 135	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))))))
137	108, 136	sylibr 234	. 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)}))
138		nocvxmin 27747	. 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)}))
139	45, 47, 137, 138	syl3anc 1374	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∃!wreu 3341  {crab 3390  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ∩ cint 4890   class class class wbr 5086   “ cima 5634  suc csuc 6326  ‘cfv 6499   No csur 27603   <s clts 27604   bday cbday 27605   <<s cslts 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6327  df-on 6328  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750

This theorem is referenced by:  cutcuts  27773  cutbday  27776  eqcuts  27777  cutsun12  27782  cutbdaylt  27790