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Theorem conway 32822
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 32815 . . . . 5 (𝐴 <<s 𝐵𝐴 No )
2 ssltex1 32813 . . . . 5 (𝐴 <<s 𝐵𝐴 ∈ V)
3 ssltss2 32816 . . . . 5 (𝐴 <<s 𝐵𝐵 No )
4 ssltex2 32814 . . . . 5 (𝐴 <<s 𝐵𝐵 ∈ V)
5 ssltsep 32817 . . . . 5 (𝐴 <<s 𝐵 → ∀𝑝𝐴𝑞𝐵 𝑝 <s 𝑞)
6 noeta 32780 . . . . 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 1362 . . . 4 (𝐴 <<s 𝐵 → ∃𝑦 No (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))))
8 3simpa 1129 . . . . . 6 ((∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞 ∧ ( bday 𝑦) ⊆ suc ( bday “ (𝐴𝐵))) → (∀𝑝𝐴 𝑝 <s 𝑦 ∧ ∀𝑞𝐵 𝑦 <s 𝑞))
92ad2antrr 714 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 ∈ V)
10 snex 5184 . . . . . . . . . 10 {𝑦} ∈ V
119, 10jctir 513 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → (𝐴 ∈ V ∧ {𝑦} ∈ V))
121ad2antrr 714 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 No )
13 snssi 4611 . . . . . . . . . . . 12 (𝑦 No → {𝑦} ⊆ No )
1413adantl 474 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝑦 No ) → {𝑦} ⊆ No )
1514adantr 473 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → {𝑦} ⊆ No )
16 vex 3411 . . . . . . . . . . . . . 14 𝑦 ∈ V
17 breq2 4929 . . . . . . . . . . . . . 14 (𝑞 = 𝑦 → (𝑝 <s 𝑞𝑝 <s 𝑦))
1816, 17ralsn 4489 . . . . . . . . . . . . 13 (∀𝑞 ∈ {𝑦}𝑝 <s 𝑞𝑝 <s 𝑦)
1918ralbii 3108 . . . . . . . . . . . 12 (∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑝𝐴 𝑝 <s 𝑦)
2019biimpri 220 . . . . . . . . . . 11 (∀𝑝𝐴 𝑝 <s 𝑦 → ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)
2120adantl 474 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)
2212, 15, 213jca 1109 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → (𝐴 No ∧ {𝑦} ⊆ No ∧ ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞))
23 brsslt 32812 . . . . . . . . 9 (𝐴 <<s {𝑦} ↔ ((𝐴 ∈ V ∧ {𝑦} ∈ V) ∧ (𝐴 No ∧ {𝑦} ⊆ No ∧ ∀𝑝𝐴𝑞 ∈ {𝑦}𝑝 <s 𝑞)))
2411, 22, 23sylanbrc 575 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑝𝐴 𝑝 <s 𝑦) → 𝐴 <<s {𝑦})
2524ex 405 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑝𝐴 𝑝 <s 𝑦𝐴 <<s {𝑦}))
264ad2antrr 714 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 ∈ V)
2726, 10jctil 512 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ({𝑦} ∈ V ∧ 𝐵 ∈ V))
2814adantr 473 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} ⊆ No )
293ad2antrr 714 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → 𝐵 No )
30 ralcom 3288 . . . . . . . . . . . 12 (∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞 ↔ ∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞)
31 breq1 4928 . . . . . . . . . . . . . 14 (𝑝 = 𝑦 → (𝑝 <s 𝑞𝑦 <s 𝑞))
3216, 31ralsn 4489 . . . . . . . . . . . . 13 (∀𝑝 ∈ {𝑦}𝑝 <s 𝑞𝑦 <s 𝑞)
3332ralbii 3108 . . . . . . . . . . . 12 (∀𝑞𝐵𝑝 ∈ {𝑦}𝑝 <s 𝑞 ↔ ∀𝑞𝐵 𝑦 <s 𝑞)
3430, 33sylbbr 228 . . . . . . . . . . 11 (∀𝑞𝐵 𝑦 <s 𝑞 → ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)
3534adantl 474 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)
3628, 29, 353jca 1109 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → ({𝑦} ⊆ No 𝐵 No ∧ ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞))
37 brsslt 32812 . . . . . . . . 9 ({𝑦} <<s 𝐵 ↔ (({𝑦} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑦} ⊆ No 𝐵 No ∧ ∀𝑝 ∈ {𝑦}∀𝑞𝐵 𝑝 <s 𝑞)))
3827, 36, 37sylanbrc 575 . . . . . . . 8 (((𝐴 <<s 𝐵𝑦 No ) ∧ ∀𝑞𝐵 𝑦 <s 𝑞) → {𝑦} <<s 𝐵)
3938ex 405 . . . . . . 7 ((𝐴 <<s 𝐵𝑦 No ) → (∀𝑞𝐵 𝑦 <s 𝑞 → {𝑦} <<s 𝐵))
4025, 39anim12d 600 . . . . . 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 3212 . . . 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 4219 . . 3 ({𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅ ↔ ∃𝑦 No (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵))
4543, 44sylibr 226 . 2 (𝐴 <<s 𝐵 → {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ≠ ∅)
46 ssrab2 3939 . . 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 1198 . . . . . 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 714 . . . . . . . 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 5184 . . . . . . . 8 {𝑟} ∈ V
5149, 50jctir 513 . . . . . . 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 714 . . . . . . . 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 4611 . . . . . . . . 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 3851 . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 473 . . . . . . . . . . 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 763 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → 𝐴 <<s {𝑝})
6059adantl 474 . . . . . . . . . . . . . 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 32817 . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . 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 3411 . . . . . . . . . . . . 13 𝑝 ∈ V
65 breq2 4929 . . . . . . . . . . . . 13 (𝑦 = 𝑝 → (𝑥 <s 𝑦𝑥 <s 𝑝))
6664, 65ralsn 4489 . . . . . . . . . . . 12 (∀𝑦 ∈ {𝑝}𝑥 <s 𝑦𝑥 <s 𝑝)
6763, 66sylib 210 . . . . . . . . . . 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 769 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 32796 . . . . . . . . . 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 3411 . . . . . . . . . . 11 𝑟 ∈ V
72 breq2 4929 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝑥 <s 𝑦𝑥 <s 𝑟))
7371, 72ralsn 4489 . . . . . . . . . 10 (∀𝑦 ∈ {𝑟}𝑥 <s 𝑦𝑥 <s 𝑟)
7470, 73sylibr 226 . . . . . . . . 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 3125 . . . . . . . 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 1109 . . . . . . 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 32812 . . . . . . 7 (𝐴 <<s {𝑟} ↔ ((𝐴 ∈ V ∧ {𝑟} ∈ V) ∧ (𝐴 No ∧ {𝑟} ⊆ No ∧ ∀𝑥𝐴𝑦 ∈ {𝑟}𝑥 <s 𝑦)))
7851, 76, 77sylanbrc 575 . . . . . 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 714 . . . . . . . 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 512 . . . . . . 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 714 . . . . . . . 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 473 . . . . . . . . . . 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 1197 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 3851 . . . . . . . . . . 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 770 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 766 . . . . . . . . . . . . . . 15 ((((𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵) ∧ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)) ∧ (𝑝 <s 𝑟𝑟 <s 𝑞)) → {𝑞} <<s 𝐵)
8988adantl 474 . . . . . . . . . . . . . 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 32817 . . . . . . . . . . . . . 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 3411 . . . . . . . . . . . . . 14 𝑞 ∈ V
93 breq1 4928 . . . . . . . . . . . . . . 15 (𝑥 = 𝑞 → (𝑥 <s 𝑦𝑞 <s 𝑦))
9493ralbidv 3140 . . . . . . . . . . . . . 14 (𝑥 = 𝑞 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦))
9592, 94ralsn 4489 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑞}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑞 <s 𝑦)
9691, 95sylib 210 . . . . . . . . . . . 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 3151 . . . . . . . . . . 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 32796 . . . . . . . . . 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 3125 . . . . . . . . 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 4928 . . . . . . . . . . 11 (𝑥 = 𝑟 → (𝑥 <s 𝑦𝑟 <s 𝑦))
101100ralbidv 3140 . . . . . . . . . 10 (𝑥 = 𝑟 → (∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦))
10271, 101ralsn 4489 . . . . . . . . 9 (∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦 ↔ ∀𝑦𝐵 𝑟 <s 𝑦)
10399, 102sylibr 226 . . . . . . . 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 1109 . . . . . . 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 32812 . . . . . . 7 ({𝑟} <<s 𝐵 ↔ (({𝑟} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑟} ⊆ No 𝐵 No ∧ ∀𝑥 ∈ {𝑟}∀𝑦𝐵 𝑥 <s 𝑦)))
10680, 104, 105sylanbrc 575 . . . . . 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 508 . . . . 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 430 . . . 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 3135 . . 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 4445 . . . . . . 7 (𝑦 = 𝑝 → {𝑦} = {𝑝})
111110breq2d 4937 . . . . . 6 (𝑦 = 𝑝 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑝}))
112110breq1d 4935 . . . . . 6 (𝑦 = 𝑝 → ({𝑦} <<s 𝐵 ↔ {𝑝} <<s 𝐵))
113111, 112anbi12d 622 . . . . 5 (𝑦 = 𝑝 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑝} ∧ {𝑝} <<s 𝐵)))
114113ralrab 3594 . . . 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 4445 . . . . . . . . 9 (𝑦 = 𝑞 → {𝑦} = {𝑞})
116115breq2d 4937 . . . . . . . 8 (𝑦 = 𝑞 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑞}))
117115breq1d 4935 . . . . . . . 8 (𝑦 = 𝑞 → ({𝑦} <<s 𝐵 ↔ {𝑞} <<s 𝐵))
118116, 117anbi12d 622 . . . . . . 7 (𝑦 = 𝑞 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑞} ∧ {𝑞} <<s 𝐵)))
119118ralrab 3594 . . . . . 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 4445 . . . . . . . . . . . 12 (𝑦 = 𝑟 → {𝑦} = {𝑟})
121120breq2d 4937 . . . . . . . . . . 11 (𝑦 = 𝑟 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑟}))
122120breq1d 4935 . . . . . . . . . . 11 (𝑦 = 𝑟 → ({𝑦} <<s 𝐵 ↔ {𝑟} <<s 𝐵))
123121, 122anbi12d 622 . . . . . . . . . 10 (𝑦 = 𝑟 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
124123elrab 3588 . . . . . . . . 9 (𝑟 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑟 No ∧ (𝐴 <<s {𝑟} ∧ {𝑟} <<s 𝐵)))
125124imbi2i 328 . . . . . . . 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 3108 . . . . . . 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 3108 . . . . . 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 3118 . . . . . . 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 3108 . . . . . 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 295 . . . . 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 3108 . . . 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 3118 . . . . . . 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 3108 . . . . . 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 3118 . . . . . 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 267 . . . . 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 3108 . . . 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 295 . . 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 226 . 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 32806 . 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 1352 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 387  w3a 1069   = wceq 1508  wcel 2051  wne 2960  wral 3081  wrex 3082  ∃!wreu 3083  {crab 3085  Vcvv 3408  cun 3820  wss 3822  c0 4172  {csn 4435   cuni 4708   cint 4745   class class class wbr 4925  cima 5406  suc csuc 6028  cfv 6185   No csur 32705   <s cslt 32706   bday cbday 32707   <<s csslt 32808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-1o 7903  df-2o 7904  df-no 32708  df-slt 32709  df-bday 32710  df-sslt 32809
This theorem is referenced by:  scutcut  32824  scutbday  32825  scutun12  32829  scutbdaylt  32834
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