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Theorem madebdaylemlrcut 27393
Description: Lemma for madebday 27394. If the inductive hypothesis of madebday 27394 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27397 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 27365 . . 3 (𝑋 No → ( L ‘𝑋) <<s {𝑋})
21adantl 483 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( L ‘𝑋) <<s {𝑋})
3 ssltright 27366 . . 3 (𝑋 No → {𝑋} <<s ( R ‘𝑋))
43adantl 483 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → {𝑋} <<s ( R ‘𝑋))
5 fveq2 6892 . . . . . . . . 9 (𝑋 = 𝑤 → ( bday 𝑋) = ( bday 𝑤))
6 eqimss 4041 . . . . . . . . 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 27292 . . . . . . . . . 10 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10 vex 3479 . . . . . . . . . . . 12 𝑤 ∈ V
11 breq2 5153 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑥 <s 𝑦𝑥 <s 𝑤))
1210, 11ralsn 4686 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦𝑥 <s 𝑤)
1312ralbii 3094 . . . . . . . . . 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 27292 . . . . . . . . . 10 ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16 breq1 5152 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦 <s 𝑥𝑤 <s 𝑥))
1716ralbidv 3178 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
1810, 17ralsn 4686 . . . . . . . . . 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 614 . . . . . . . 8 ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21 leftval 27358 . . . . . . . . . . . . . . 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 3326 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24 rightval 27359 . . . . . . . . . . . . . . 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 3326 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
2723, 26anbi12d 632 . . . . . . . . . . . 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 5152 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 <s 𝑋𝑥 <s 𝑋))
2928ralrab 3690 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤))
30 breq2 5153 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑋 <s 𝑧𝑋 <s 𝑥))
3130ralrab 3690 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))
3229, 31anbi12i 628 . . . . . . . . . . . 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 287 . . . . . . . . . . 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 726 . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 27249 . . . . . . . . . . . . . 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 27278 . . . . . . . . . . . . . . . 16 ( bday 𝑋) ∈ On
39 bdayelon 27278 . . . . . . . . . . . . . . . 16 ( bday 𝑤) ∈ On
40 ontri1 6399 . . . . . . . . . . . . . . . 16 ((( bday 𝑋) ∈ On ∧ ( bday 𝑤) ∈ On) → (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋)))
4138, 39, 40mp2an 691 . . . . . . . . . . . . . . 15 (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋))
4241con2bii 358 . . . . . . . . . . . . . 14 (( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑋) ⊆ ( bday 𝑤))
43 simplll 774 . . . . . . . . . . . . . . . 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 27392 . . . . . . . . . . . . . . . 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 1467 . . . . . . . . . . . . . . 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 27254 . . . . . . . . . . . . . . . . . 18 ((𝑋 No 𝑤 No ) → (𝑋𝑤 ↔ (𝑋 <s 𝑤𝑤 <s 𝑋)))
4746ad2ant2lr 747 . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑋 <s 𝑥𝑋 <s 𝑤))
50 breq2 5153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑤 <s 𝑥𝑤 <s 𝑤))
5149, 50imbi12d 345 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑋 <s 𝑥𝑤 <s 𝑥) ↔ (𝑋 <s 𝑤𝑤 <s 𝑤)))
5251rspccv 3610 . . . . . . . . . . . . . . . . . . . 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 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 𝑋𝑥 <s 𝑤))
56 breq1 5152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑋𝑤 <s 𝑋))
57 breq1 5152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑤𝑤 <s 𝑤))
5856, 57imbi12d 345 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑥 <s 𝑋𝑥 <s 𝑤) ↔ (𝑤 <s 𝑋𝑤 <s 𝑤)))
5958rspccv 3610 . . . . . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . 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 458 . . . . . . . . . 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 456 . . . . . . . 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 682 . . . . . . 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 3029 . . . . . 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 458 . . . . 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 3147 . . . 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 27275 . . . . . 6 bday Fn No
77 ssrab2 4078 . . . . . 6 {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78 fnssintima 7359 . . . . . 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 691 . . . . 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 4639 . . . . . . . 8 (𝑧 = 𝑤 → {𝑧} = {𝑤})
8180breq2d 5161 . . . . . . 7 (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
8280breq1d 5159 . . . . . . 7 (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
8381, 82anbi12d 632 . . . . . 6 (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
8483ralrab 3690 . . . . 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 275 . . . 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 4639 . . . . . . . 8 (𝑧 = 𝑋 → {𝑧} = {𝑋})
8887breq2d 5161 . . . . . . 7 (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
8987breq1d 5159 . . . . . . 7 (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
9088, 89anbi12d 632 . . . . . 6 (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
91 simpr 486 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 No )
922, 4jca 513 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋)))
9390, 91, 92elrabd 3686 . . . . 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 7235 . . . . 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 1466 . . . 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 4968 . . . 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 4000 . 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 27367 . . . 4 ( L ‘𝑋) <<s ( R ‘𝑋)
100 eqscut 27306 . . . 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, 100mpan 689 . . 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 483 . 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 1343 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  {crab 3433  wss 3949  {csn 4629   cint 4951   class class class wbr 5149  cima 5680  Oncon0 6365   Fn wfn 6539  cfv 6544  (class class class)co 7409   No csur 27143   <s cslt 27144   bday cbday 27145   <<s csslt 27282   |s cscut 27284   M cmade 27337   O cold 27338   L cleft 27340   R cright 27341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285  df-made 27342  df-old 27343  df-left 27345  df-right 27346
This theorem is referenced by:  madebday  27394  lrcut  27397
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