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Theorem conway 27778
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.
StepHypRef Expression
1 ssltss1 27767 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
2 ssltex1 27765 . . . . 5 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltss2 27768 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
4 ssltex2 27766 . . . . 5 (𝐴 <<s 𝐵𝐵 ∈ V)
5 ssltsep 27769 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑝𝐴𝑞𝐵 𝑝 <s 𝑞)
6 noeta2 27763 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑝𝐴𝑞𝐵 𝑝 <s 𝑞) → ∃𝑦 No (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))))
71, 2, 3, 4, 5, 6syl221anc 1378 . . . 4 (𝐴 <<s 𝐵 → ∃𝑦 No (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))))
8 3simpa 1145 . . . . . 6 ((∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))) → (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞))
92ad2antrr 724 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10 vsnex 5431 . . . . . . . . . 10 {𝑦} ∈ V
119, 10jctir 519 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
121ad2antrr 724 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 No )
13 snssi 4813 . . . . . . . . . . . 12 (𝑦 No → {𝑦} ⊆ No )
1413adantl 480 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝑦 No ) → {𝑦} ⊆ No )
1514adantr 479 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16 vex 3465 . . . . . . . . . . . . . 14 𝑦 ∈ V
17 breq2 5153 . . . . . . . . . . . . . 14 (𝑞 = 𝑦 → (𝑝 <s 𝑞𝑝 <s 𝑦))
1816, 17ralsn 4687 . . . . . . . . . . . . 13 (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞𝑝 <s 𝑦)
1918ralbii 3082 . . . . . . . . . . . 12 (∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝𝐴 𝑝 <s 𝑦)
2019biimpri 227 . . . . . . . . . . 11 (∀𝑝𝐴 𝑝 <s 𝑦 → ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)
2120adantl 480 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)
2212, 15, 213jca 1125 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → (𝐴 No ∧ {𝑦} ⊆ No ∧ ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞))
23 brsslt 27764 . . . . . . . . 9 (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 No ∧ {𝑦} ⊆ No ∧ ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
2411, 22, 23sylanbrc 581 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
2524ex 411 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑝𝐴 𝑝 <s 𝑦𝐴 <<s {𝑦}))
264ad2antrr 724 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
2726, 10jctil 518 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
2814adantr 479 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
293ad2antrr 724 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 No )
30 ralcom 3276 . . . . . . . . . . . 12 (∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞 ↔ ∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31 breq1 5152 . . . . . . . . . . . . . 14 (𝑝 = 𝑦 → (𝑝 <s 𝑞𝑦 <s 𝑞))
3216, 31ralsn 4687 . . . . . . . . . . . . 13 (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞𝑦 <s 𝑞)
3332ralbii 3082 . . . . . . . . . . . 12 (∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞𝐵 𝑦 <s 𝑞)
3430, 33sylbbr 235 . . . . . . . . . . 11 (∀𝑞𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)
3534adantl 480 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)
3628, 29, 353jca 1125 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ({𝑦} ⊆ No 𝐵 No ∧ ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞))
37 brsslt 27764 . . . . . . . . 9 ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No 𝐵 No ∧ ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)))
3827, 36, 37sylanbrc 581 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
3938ex 411 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑞𝐵 𝑦 <s 𝑞 → {𝑦} <<s 𝐵))
4025, 39anim12d 607 . . . . . 6 ((𝐴 <<s 𝐵𝑦 No ) → ((∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
418, 40syl5 34 . . . . 5 ((𝐴 <<s 𝐵𝑦 No ) → ((∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))) → (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
4241reximdva 3157 . . . 4 (𝐴 <<s 𝐵 → (∃𝑦 No (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))) → ∃𝑦 No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
437, 42mpd 15 . . 3 (𝐴 <<s 𝐵 → ∃𝑦 No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
44 rabn0 4387 . . 3 ({𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
4543, 44sylibr 233 . 2 (𝐴 <<s 𝐵 → {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46 ssrab2 4073 . . 3 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
4746a1i 11 . 2 (𝐴 <<s 𝐵 → {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No )
48 simplr3 1214 . . . . . 6 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → 𝑟 No )
492ad2antrr 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 𝑟𝑟 <s 𝑞))) → 𝐴 ∈ V)
50 vsnex 5431 . . . . . . . 8 {𝑟} ∈ V
5149, 50jctir 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 𝑟𝑟 <s 𝑞))) → (𝐴 ∈ V ∧ {𝑟} ∈ V))
521ad2antrr 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 𝑟𝑟 <s 𝑞))) → 𝐴 No )
53 snssi 4813 . . . . . . . . 9 (𝑟 No → {𝑟} ⊆ No )
5448, 53syl 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 𝑟𝑟 <s 𝑞))) → {𝑟} ⊆ No )
5552sselda 3976 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 𝑥 No )
56 simplr1 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 𝑟𝑟 <s 𝑞))) → 𝑝 No )
5756adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 𝑝 No )
5848adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 𝑟 No )
59 simplll 773 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → 𝐴 <<s {𝑝})
6059adantl 480 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → 𝐴 <<s {𝑝})
61 ssltsep 27769 . . . . . . . . . . . . . 14 (𝐴 <<s {𝑝} → ∀𝑥𝐴𝑦 ∈ {𝑝}𝑥 <s 𝑦)
6260, 61syl 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 𝑟𝑟 <s 𝑞))) → ∀𝑥𝐴𝑦 ∈ {𝑝}𝑥 <s 𝑦)
6362r19.21bi 3238 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → ∀𝑦 ∈ {𝑝}𝑥 <s 𝑦)
64 vex 3465 . . . . . . . . . . . . 13 𝑝 ∈ V
65 breq2 5153 . . . . . . . . . . . . 13 (𝑦 = 𝑝 → (𝑥 <s 𝑦𝑥 <s 𝑝))
6664, 65ralsn 4687 . . . . . . . . . . . 12 (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦𝑥 <s 𝑝)
6763, 66sylib 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 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 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 𝑟𝑟 <s 𝑞))) → 𝑝 <s 𝑟)
6968adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 𝑝 <s 𝑟)
7055, 57, 58, 67, 69slttrd 27738 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → 𝑥 <s 𝑟)
71 vex 3465 . . . . . . . . . . 11 𝑟 ∈ V
72 breq2 5153 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝑥 <s 𝑦𝑥 <s 𝑟))
7371, 72ralsn 4687 . . . . . . . . . 10 (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦𝑥 <s 𝑟)
7470, 73sylibr 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 𝑟𝑟 <s 𝑞))) ∧ 𝑥𝐴) → ∀𝑦 ∈ {𝑟}𝑥 <s 𝑦)
7574ralrimiva 3135 . . . . . . . 8 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → ∀𝑥𝐴𝑦 ∈ {𝑟}𝑥 <s 𝑦)
7652, 54, 753jca 1125 . . . . . . 7 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → (𝐴 No ∧ {𝑟} ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ {𝑟}𝑥 <s 𝑦))
77 brsslt 27764 . . . . . . 7 (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 No ∧ {𝑟} ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
7851, 76, 77sylanbrc 581 . . . . . 6 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → 𝐴 <<s {𝑟})
794ad2antrr 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 𝑟𝑟 <s 𝑞))) → 𝐵 ∈ V)
8079, 50jctil 518 . . . . . . 7 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → ({𝑟} ∈ V ∧ 𝐵 ∈ V))
813ad2antrr 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 𝑟𝑟 <s 𝑞))) → 𝐵 No )
8248adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 𝑟 No )
83 simplr2 1213 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → 𝑞 No )
8483adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 𝑞 No )
8581sselda 3976 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 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 𝑟𝑟 <s 𝑞))) → 𝑟 <s 𝑞)
8786adantr 479 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 𝑟 <s 𝑞)
88 simplrr 776 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → {𝑞} <<s 𝐵)
8988adantl 480 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → {𝑞} <<s 𝐵)
90 ssltsep 27769 . . . . . . . . . . . . . 14 ({𝑞} <<s 𝐵 → ∀𝑥 ∈ {𝑞}∀𝑦𝐵 𝑥 <s 𝑦)
9189, 90syl 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 𝑟𝑟 <s 𝑞))) → ∀𝑥 ∈ {𝑞}∀𝑦𝐵 𝑥 <s 𝑦)
92 vex 3465 . . . . . . . . . . . . . 14 𝑞 ∈ V
93 breq1 5152 . . . . . . . . . . . . . . 15 (𝑥 = 𝑞 → (𝑥 <s 𝑦𝑞 <s 𝑦))
9493ralbidv 3167 . . . . . . . . . . . . . 14 (𝑥 = 𝑞 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦))
9592, 94ralsn 4687 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑞}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦)
9691, 95sylib 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 𝑟𝑟 <s 𝑞))) → ∀𝑦𝐵 𝑞 <s 𝑦)
9796r19.21bi 3238 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 𝑞 <s 𝑦)
9882, 84, 85, 87, 97slttrd 27738 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) ∧ 𝑦𝐵) → 𝑟 <s 𝑦)
9998ralrimiva 3135 . . . . . . . . 9 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → ∀𝑦𝐵 𝑟 <s 𝑦)
100 breq1 5152 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 <s 𝑦𝑟 <s 𝑦))
101100ralbidv 3167 . . . . . . . . . 10 (𝑥 = 𝑟 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦))
10271, 101ralsn 4687 . . . . . . . . 9 (∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦)
10399, 102sylibr 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 𝑟𝑟 <s 𝑞))) → ∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦)
10454, 81, 1033jca 1125 . . . . . . 7 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → ({𝑟} ⊆ No 𝐵 No ∧ ∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦))
105 brsslt 27764 . . . . . . 7 ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No 𝐵 No ∧ ∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦)))
10680, 104, 105sylanbrc 581 . . . . . 6 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → {𝑟} <<s 𝐵)
10748, 78, 106jca32 514 . . . . 5 (((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) ∧ (((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞))) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
108107exp44 436 . . . 4 ((𝐴 <<s 𝐵 ∧ (𝑝 No 𝑞 No 𝑟 No )) → ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
109108ralrimivvva 3193 . . 3 (𝐴 <<s 𝐵 → ∀𝑝 No 𝑞 No 𝑟 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
110 sneq 4640 . . . . . . 7 (𝑦 = 𝑝 → {𝑦} = {𝑝})
111110breq2d 5161 . . . . . 6 (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
112110breq1d 5159 . . . . . 6 (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
113111, 112anbi12d 630 . . . . 5 (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
114113ralrab 3685 . . . 4 (∀𝑝 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 No 𝑟 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
115 sneq 4640 . . . . . . . . 9 (𝑦 = 𝑞 → {𝑦} = {𝑞})
116115breq2d 5161 . . . . . . . 8 (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
117115breq1d 5159 . . . . . . . 8 (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
118116, 117anbi12d 630 . . . . . . 7 (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
119118ralrab 3685 . . . . . 6 (∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))) ↔ ∀𝑞 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
120 sneq 4640 . . . . . . . . . . . 12 (𝑦 = 𝑟 → {𝑦} = {𝑟})
121120breq2d 5161 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
122120breq1d 5159 . . . . . . . . . . 11 (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
123121, 122anbi12d 630 . . . . . . . . . 10 (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124123elrab 3679 . . . . . . . . 9 (𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
125124imbi2i 335 . . . . . . . 8 (((𝑝 <s 𝑟𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
126125ralbii 3082 . . . . . . 7 (∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
127126ralbii 3082 . . . . . 6 (∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))
128 r19.21v 3169 . . . . . . 7 (∀𝑟 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))) ↔ ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
129128ralbii 3082 . . . . . 6 (∀𝑞 No 𝑟 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))) ↔ ∀𝑞 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
130119, 127, 1293bitr4i 302 . . . . 5 (∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ∀𝑞 No 𝑟 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
131130ralbii 3082 . . . 4 (∀𝑝 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))))
132 r19.21v 3169 . . . . . . 7 (∀𝑟 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
133132ralbii 3082 . . . . . 6 (∀𝑞 No 𝑟 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
134 r19.21v 3169 . . . . . 6 (∀𝑞 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ∀𝑟 No ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
135133, 134bitri 274 . . . . 5 (∀𝑞 No 𝑟 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
136135ralbii 3082 . . . 4 (∀𝑝 No 𝑞 No 𝑟 No ((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) → ((𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵) → ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
137114, 131, 1363bitr4i 302 . . 3 (∀𝑝 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑞 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}∀𝑟 No ((𝑝 <s 𝑟𝑟 <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 𝑟𝑟 <s 𝑞) → (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵))))))
138109, 137sylibr 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 𝑟𝑟 <s 𝑞) → 𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
139 nocvxmin 27757 . 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 𝑟𝑟 <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)}))
14045, 47, 138, 139syl3anc 1368 1 (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  ∃!wreu 3361  {crab 3418  Vcvv 3461  cun 3942  wss 3944  c0 4322  {csn 4630   cuni 4909   cint 4950   class class class wbr 5149  cima 5681  suc csuc 6373  cfv 6549   No csur 27618   <s cslt 27619   bday cbday 27620   <<s csslt 27759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-1o 8487  df-2o 8488  df-no 27621  df-slt 27622  df-bday 27623  df-sslt 27760
This theorem is referenced by:  scutcut  27780  scutbday  27783  eqscut  27784  scutun12  27789  scutbdaylt  27797
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