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Theorem madebdaylemlrcut 27228
Description: Lemma for madebday 27229. If the inductive hypothesis of madebday 27229 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27232 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
Assertion
Ref Expression
madebdaylemlrcut ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Distinct variable group:   𝑦,𝑏,𝑋

Proof of Theorem madebdaylemlrcut
Dummy variables 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssltleft 27200 . . 3 (𝑋 No → ( L ‘𝑋) <<s {𝑋})
21adantl 482 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( L ‘𝑋) <<s {𝑋})
3 ssltright 27201 . . 3 (𝑋 No → {𝑋} <<s ( R ‘𝑋))
43adantl 482 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → {𝑋} <<s ( R ‘𝑋))
5 fveq2 6842 . . . . . . . . 9 (𝑋 = 𝑤 → ( bday 𝑋) = ( bday 𝑤))
6 eqimss 4000 . . . . . . . . 9 (( bday 𝑋) = ( bday 𝑤) → ( bday 𝑋) ⊆ ( bday 𝑤))
75, 6syl 17 . . . . . . . 8 (𝑋 = 𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤))
87a1i 11 . . . . . . 7 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → (𝑋 = 𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤)))
9 ssltsep 27130 . . . . . . . . . 10 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10 vex 3449 . . . . . . . . . . . 12 𝑤 ∈ V
11 breq2 5109 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑥 <s 𝑦𝑥 <s 𝑤))
1210, 11ralsn 4642 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦𝑥 <s 𝑤)
1312ralbii 3096 . . . . . . . . . 10 (∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
149, 13sylib 217 . . . . . . . . 9 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
15 ssltsep 27130 . . . . . . . . . 10 ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16 breq1 5108 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦 <s 𝑥𝑤 <s 𝑥))
1716ralbidv 3174 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
1810, 17ralsn 4642 . . . . . . . . . 10 (∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
1915, 18sylib 217 . . . . . . . . 9 ({𝑤} <<s ( R ‘𝑋) → ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
2014, 19anim12i 613 . . . . . . . 8 ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21 leftval 27193 . . . . . . . . . . . . . . 15 ( L ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}
2221a1i 11 . . . . . . . . . . . . . 14 (𝑋 No → ( L ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋})
2322raleqdv 3313 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24 rightval 27194 . . . . . . . . . . . . . . 15 ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}
2524a1i 11 . . . . . . . . . . . . . 14 (𝑋 No → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧})
2625raleqdv 3313 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
2723, 26anbi12d 631 . . . . . . . . . . . 12 (𝑋 No → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) ↔ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ∧ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥)))
28 breq1 5108 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 <s 𝑋𝑥 <s 𝑋))
2928ralrab 3651 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤))
30 breq2 5109 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑋 <s 𝑧𝑋 <s 𝑥))
3130ralrab 3651 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))
3229, 31anbi12i 627 . . . . . . . . . . . 12 ((∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ∧ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥) ↔ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))
3327, 32bitrdi 286 . . . . . . . . . . 11 (𝑋 No → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) ↔ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))))
3433ad2antlr 725 . . . . . . . . . 10 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ 𝑤 No ) → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) ↔ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))))
35 simplrl 775 . . . . . . . . . . . . . 14 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → 𝑤 No )
36 sltirr 27094 . . . . . . . . . . . . . 14 (𝑤 No → ¬ 𝑤 <s 𝑤)
3735, 36syl 17 . . . . . . . . . . . . 13 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → ¬ 𝑤 <s 𝑤)
38 bdayelon 27116 . . . . . . . . . . . . . . . 16 ( bday 𝑋) ∈ On
39 bdayelon 27116 . . . . . . . . . . . . . . . 16 ( bday 𝑤) ∈ On
40 ontri1 6351 . . . . . . . . . . . . . . . 16 ((( bday 𝑋) ∈ On ∧ ( bday 𝑤) ∈ On) → (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋)))
4138, 39, 40mp2an 690 . . . . . . . . . . . . . . 15 (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋))
4241con2bii 357 . . . . . . . . . . . . . 14 (( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑋) ⊆ ( bday 𝑤))
43 simplll 773 . . . . . . . . . . . . . . . 16 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → ∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)))
44 madebdaylemold 27227 . . . . . . . . . . . . . . . 16 ((( bday 𝑋) ∈ On ∧ ∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑤 No ) → (( bday 𝑤) ∈ ( bday 𝑋) → 𝑤 ∈ ( O ‘( bday 𝑋))))
4538, 43, 35, 44mp3an2i 1466 . . . . . . . . . . . . . . 15 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → (( bday 𝑤) ∈ ( bday 𝑋) → 𝑤 ∈ ( O ‘( bday 𝑋))))
46 slttrine 27099 . . . . . . . . . . . . . . . . . 18 ((𝑋 No 𝑤 No ) → (𝑋𝑤 ↔ (𝑋 <s 𝑤𝑤 <s 𝑋)))
4746ad2ant2lr 746 . . . . . . . . . . . . . . . . 17 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑋𝑤 ↔ (𝑋 <s 𝑤𝑤 <s 𝑋)))
48 simprrr 780 . . . . . . . . . . . . . . . . . . . 20 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))
49 breq2 5109 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑋 <s 𝑥𝑋 <s 𝑤))
50 breq2 5109 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑤 <s 𝑥𝑤 <s 𝑤))
5149, 50imbi12d 344 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑋 <s 𝑥𝑤 <s 𝑥) ↔ (𝑋 <s 𝑤𝑤 <s 𝑤)))
5251rspccv 3578 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑋 <s 𝑤𝑤 <s 𝑤)))
5348, 52syl 17 . . . . . . . . . . . . . . . . . . 19 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑋 <s 𝑤𝑤 <s 𝑤)))
5453com23 86 . . . . . . . . . . . . . . . . . 18 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑋 <s 𝑤 → (𝑤 ∈ ( O ‘( bday 𝑋)) → 𝑤 <s 𝑤)))
55 simprrl 779 . . . . . . . . . . . . . . . . . . . 20 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤))
56 breq1 5108 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑋𝑤 <s 𝑋))
57 breq1 5108 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑤𝑤 <s 𝑤))
5856, 57imbi12d 344 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑥 <s 𝑋𝑥 <s 𝑤) ↔ (𝑤 <s 𝑋𝑤 <s 𝑤)))
5958rspccv 3578 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑤 <s 𝑋𝑤 <s 𝑤)))
6055, 59syl 17 . . . . . . . . . . . . . . . . . . 19 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑤 <s 𝑋𝑤 <s 𝑤)))
6160com23 86 . . . . . . . . . . . . . . . . . 18 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑤 <s 𝑋 → (𝑤 ∈ ( O ‘( bday 𝑋)) → 𝑤 <s 𝑤)))
6254, 61jaod 857 . . . . . . . . . . . . . . . . 17 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → ((𝑋 <s 𝑤𝑤 <s 𝑋) → (𝑤 ∈ ( O ‘( bday 𝑋)) → 𝑤 <s 𝑤)))
6347, 62sylbid 239 . . . . . . . . . . . . . . . 16 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑋𝑤 → (𝑤 ∈ ( O ‘( bday 𝑋)) → 𝑤 <s 𝑤)))
6463imp 407 . . . . . . . . . . . . . . 15 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → (𝑤 ∈ ( O ‘( bday 𝑋)) → 𝑤 <s 𝑤))
6545, 64syld 47 . . . . . . . . . . . . . 14 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → (( bday 𝑤) ∈ ( bday 𝑋) → 𝑤 <s 𝑤))
6642, 65biimtrrid 242 . . . . . . . . . . . . 13 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → (¬ ( bday 𝑋) ⊆ ( bday 𝑤) → 𝑤 <s 𝑤))
6737, 66mt3d 148 . . . . . . . . . . . 12 ((((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) ∧ 𝑋𝑤) → ( bday 𝑋) ⊆ ( bday 𝑤))
6867ex 413 . . . . . . . . . . 11 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)))) → (𝑋𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤)))
6968expr 457 . . . . . . . . . 10 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ 𝑤 No ) → ((∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥)) → (𝑋𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤))))
7034, 69sylbid 239 . . . . . . . . 9 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ 𝑤 No ) → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) → (𝑋𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤))))
7170impr 455 . . . . . . . 8 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))) → (𝑋𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤)))
7220, 71sylanr2 681 . . . . . . 7 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → (𝑋𝑤 → ( bday 𝑋) ⊆ ( bday 𝑤)))
738, 72pm2.61dne 3031 . . . . . 6 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ (𝑤 No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → ( bday 𝑋) ⊆ ( bday 𝑤))
7473expr 457 . . . . 5 (((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) ∧ 𝑤 No ) → ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday 𝑋) ⊆ ( bday 𝑤)))
7574ralrimiva 3143 . . . 4 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ∀𝑤 No ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday 𝑋) ⊆ ( bday 𝑤)))
76 bdayfn 27113 . . . . . 6 bday Fn No
77 ssrab2 4037 . . . . . 6 {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78 fnssintima 7307 . . . . . 6 (( bday Fn No ∧ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No ) → (( bday 𝑋) ⊆ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ↔ ∀𝑤 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ( bday 𝑋) ⊆ ( bday 𝑤)))
7976, 77, 78mp2an 690 . . . . 5 (( bday 𝑋) ⊆ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ↔ ∀𝑤 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ( bday 𝑋) ⊆ ( bday 𝑤))
80 sneq 4596 . . . . . . . 8 (𝑧 = 𝑤 → {𝑧} = {𝑤})
8180breq2d 5117 . . . . . . 7 (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
8280breq1d 5115 . . . . . . 7 (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
8381, 82anbi12d 631 . . . . . 6 (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
8483ralrab 3651 . . . . 5 (∀𝑤 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ( bday 𝑋) ⊆ ( bday 𝑤) ↔ ∀𝑤 No ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday 𝑋) ⊆ ( bday 𝑤)))
8579, 84bitri 274 . . . 4 (( bday 𝑋) ⊆ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ↔ ∀𝑤 No ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday 𝑋) ⊆ ( bday 𝑤)))
8675, 85sylibr 233 . . 3 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( bday 𝑋) ⊆ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
87 sneq 4596 . . . . . . . 8 (𝑧 = 𝑋 → {𝑧} = {𝑋})
8887breq2d 5117 . . . . . . 7 (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
8987breq1d 5115 . . . . . . 7 (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
9088, 89anbi12d 631 . . . . . 6 (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
91 simpr 485 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 No )
922, 4jca 512 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋)))
9390, 91, 92elrabd 3647 . . . . 5 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))})
94 fnfvima 7183 . . . . 5 (( bday Fn No ∧ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No 𝑋 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) → ( bday 𝑋) ∈ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
9576, 77, 93, 94mp3an12i 1465 . . . 4 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( bday 𝑋) ∈ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
96 intss1 4924 . . . 4 (( bday 𝑋) ∈ ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) → ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ⊆ ( bday 𝑋))
9795, 96syl 17 . . 3 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ⊆ ( bday 𝑋))
9886, 97eqssd 3961 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( bday 𝑋) = ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
99 lltropt 27202 . . . 4 (𝑋 No → ( L ‘𝑋) <<s ( R ‘𝑋))
100 eqscut 27144 . . . 4 ((( L ‘𝑋) <<s ( R ‘𝑋) ∧ 𝑋 No ) → ((( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋 ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋) ∧ ( bday 𝑋) = ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))))
10199, 100mpancom 686 . . 3 (𝑋 No → ((( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋 ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋) ∧ ( bday 𝑋) = ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))))
102101adantl 482 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ((( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋 ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋) ∧ ( bday 𝑋) = ( bday “ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))))
1032, 4, 98, 102mpbir3and 1342 1 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  wss 3910  {csn 4586   cint 4907   class class class wbr 5105  cima 5636  Oncon0 6317   Fn wfn 6491  cfv 6496  (class class class)co 7357   No csur 26988   <s cslt 26989   bday cbday 26990   <<s csslt 27120   |s cscut 27122   M cmade 27172   O cold 27173   L cleft 27175   R cright 27176
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sslt 27121  df-scut 27123  df-made 27177  df-old 27178  df-left 27180  df-right 27181
This theorem is referenced by:  madebday  27229  lrcut  27232
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