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Theorem conway 33372
 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.
StepHypRef Expression
1 ssltss1 33365 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
2 ssltex1 33363 . . . . 5 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltss2 33366 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
4 ssltex2 33364 . . . . 5 (𝐴 <<s 𝐵𝐵 ∈ V)
5 ssltsep 33367 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑝𝐴𝑞𝐵 𝑝 <s 𝑞)
6 noeta 33330 . . . . 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 725 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10 snex 5300 . . . . . . . . . 10 {𝑦} ∈ V
119, 10jctir 524 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
121ad2antrr 725 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 No )
13 snssi 4704 . . . . . . . . . . . 12 (𝑦 No → {𝑦} ⊆ No )
1413adantl 485 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝑦 No ) → {𝑦} ⊆ No )
1514adantr 484 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16 vex 3447 . . . . . . . . . . . . . 14 𝑦 ∈ V
17 breq2 5037 . . . . . . . . . . . . . 14 (𝑞 = 𝑦 → (𝑝 <s 𝑞𝑝 <s 𝑦))
1816, 17ralsn 4582 . . . . . . . . . . . . 13 (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞𝑝 <s 𝑦)
1918ralbii 3136 . . . . . . . . . . . 12 (∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝𝐴 𝑝 <s 𝑦)
2019biimpri 231 . . . . . . . . . . 11 (∀𝑝𝐴 𝑝 <s 𝑦 → ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)
2120adantl 485 . . . . . . . . . 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 33362 . . . . . . . . 9 (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 No ∧ {𝑦} ⊆ No ∧ ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
2411, 22, 23sylanbrc 586 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
2524ex 416 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑝𝐴 𝑝 <s 𝑦𝐴 <<s {𝑦}))
264ad2antrr 725 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
2726, 10jctil 523 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
2814adantr 484 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
293ad2antrr 725 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 No )
30 ralcom 3310 . . . . . . . . . . . 12 (∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞 ↔ ∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31 breq1 5036 . . . . . . . . . . . . . 14 (𝑝 = 𝑦 → (𝑝 <s 𝑞𝑦 <s 𝑞))
3216, 31ralsn 4582 . . . . . . . . . . . . 13 (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞𝑦 <s 𝑞)
3332ralbii 3136 . . . . . . . . . . . 12 (∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞𝐵 𝑦 <s 𝑞)
3430, 33sylbbr 239 . . . . . . . . . . 11 (∀𝑞𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)
3534adantl 485 . . . . . . . . . 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 33362 . . . . . . . . 9 ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No 𝐵 No ∧ ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)))
3827, 36, 37sylanbrc 586 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
3938ex 416 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑞𝐵 𝑦 <s 𝑞 → {𝑦} <<s 𝐵))
4025, 39anim12d 611 . . . . . 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 3236 . . . 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 4296 . . 3 ({𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
4543, 44sylibr 237 . 2 (𝐴 <<s 𝐵 → {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46 ssrab2 4010 . . 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 725 . . . . . . . 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 snex 5300 . . . . . . . 8 {𝑟} ∈ V
5149, 50jctir 524 . . . . . . 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 725 . . . . . . . 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 4704 . . . . . . . . 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 3918 . . . . . . . . . . 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 484 . . . . . . . . . . 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 484 . . . . . . . . . . 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 774 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → 𝐴 <<s {𝑝})
6059adantl 485 . . . . . . . . . . . . . 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 33367 . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . 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 3447 . . . . . . . . . . . . 13 𝑝 ∈ V
65 breq2 5037 . . . . . . . . . . . . 13 (𝑦 = 𝑝 → (𝑥 <s 𝑦𝑥 <s 𝑝))
6664, 65ralsn 4582 . . . . . . . . . . . 12 (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦𝑥 <s 𝑝)
6763, 66sylib 221 . . . . . . . . . . 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 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))) → 𝑝 <s 𝑟)
6968adantr 484 . . . . . . . . . . 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 33346 . . . . . . . . . 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 3447 . . . . . . . . . . 11 𝑟 ∈ V
72 breq2 5037 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝑥 <s 𝑦𝑥 <s 𝑟))
7371, 72ralsn 4582 . . . . . . . . . 10 (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦𝑥 <s 𝑟)
7470, 73sylibr 237 . . . . . . . . 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 3152 . . . . . . . 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 33362 . . . . . . 7 (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 No ∧ {𝑟} ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
7851, 76, 77sylanbrc 586 . . . . . 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 725 . . . . . . . 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 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 𝑟𝑟 <s 𝑞))) → ({𝑟} ∈ V ∧ 𝐵 ∈ V))
813ad2antrr 725 . . . . . . . 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 484 . . . . . . . . . . 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 484 . . . . . . . . . . 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 3918 . . . . . . . . . . 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 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 𝑟𝑟 <s 𝑞))) → 𝑟 <s 𝑞)
8786adantr 484 . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → {𝑞} <<s 𝐵)
8988adantl 485 . . . . . . . . . . . . . 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 33367 . . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . . 14 𝑞 ∈ V
93 breq1 5036 . . . . . . . . . . . . . . 15 (𝑥 = 𝑞 → (𝑥 <s 𝑦𝑞 <s 𝑦))
9493ralbidv 3165 . . . . . . . . . . . . . 14 (𝑥 = 𝑞 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦))
9592, 94ralsn 4582 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑞}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦)
9691, 95sylib 221 . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 33346 . . . . . . . . . 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 3152 . . . . . . . . 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 5036 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 <s 𝑦𝑟 <s 𝑦))
101100ralbidv 3165 . . . . . . . . . 10 (𝑥 = 𝑟 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦))
10271, 101ralsn 4582 . . . . . . . . 9 (∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦)
10399, 102sylibr 237 . . . . . . . 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 33362 . . . . . . 7 ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No 𝐵 No ∧ ∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦)))
10680, 104, 105sylanbrc 586 . . . . . 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 519 . . . . 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 441 . . . 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 3160 . . 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 4538 . . . . . . 7 (𝑦 = 𝑝 → {𝑦} = {𝑝})
111110breq2d 5045 . . . . . 6 (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
112110breq1d 5043 . . . . . 6 (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
113111, 112anbi12d 633 . . . . 5 (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
114113ralrab 3636 . . . 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 4538 . . . . . . . . 9 (𝑦 = 𝑞 → {𝑦} = {𝑞})
116115breq2d 5045 . . . . . . . 8 (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
117115breq1d 5043 . . . . . . . 8 (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
118116, 117anbi12d 633 . . . . . . 7 (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
119118ralrab 3636 . . . . . 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 4538 . . . . . . . . . . . 12 (𝑦 = 𝑟 → {𝑦} = {𝑟})
121120breq2d 5045 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
122120breq1d 5043 . . . . . . . . . . 11 (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
123121, 122anbi12d 633 . . . . . . . . . 10 (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124123elrab 3631 . . . . . . . . 9 (𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
125124imbi2i 339 . . . . . . . 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 3136 . . . . . . 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 3136 . . . . . 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 3145 . . . . . . 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 3136 . . . . . 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 306 . . . . 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 3136 . . . 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 3145 . . . . . . 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 3136 . . . . . 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 3145 . . . . . 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 278 . . . . 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 3136 . . . 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 306 . . 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 237 . 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 33356 . 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  ∃!wreu 3111  {crab 3113  Vcvv 3444   ∪ cun 3882   ⊆ wss 3884  ∅c0 4246  {csn 4528  ∪ cuni 4803  ∩ cint 4841   class class class wbr 5033   “ cima 5526  suc csuc 6165  ‘cfv 6328   No csur 33255
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