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Theorem madebdaylemlrcut 27628
Description: Lemma for madebday 27629. If the inductive hypothesis of madebday 27629 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27632 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 27600 . . 3 (𝑋 No → ( L ‘𝑋) <<s {𝑋})
21adantl 480 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( L ‘𝑋) <<s {𝑋})
3 ssltright 27601 . . 3 (𝑋 No → {𝑋} <<s ( R ‘𝑋))
43adantl 480 . 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 27526 . . . . . . . . . 10 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10 vex 3476 . . . . . . . . . . . 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 3091 . . . . . . . . . 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 27526 . . . . . . . . . 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 3175 . . . . . . . . . . 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 611 . . . . . . . 8 ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21 leftval 27593 . . . . . . . . . . . . . . 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 3323 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24 rightval 27594 . . . . . . . . . . . . . . 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 3323 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
2723, 26anbi12d 629 . . . . . . . . . . . 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 625 . . . . . . . . . . . 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 723 . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 27483 . . . . . . . . . . . . . 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 27512 . . . . . . . . . . . . . . . 16 ( bday 𝑋) ∈ On
39 bdayelon 27512 . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . 15 (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋))
4241con2bii 356 . . . . . . . . . . . . . 14 (( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑋) ⊆ ( bday 𝑤))
43 simplll 771 . . . . . . . . . . . . . . . 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 27627 . . . . . . . . . . . . . . . 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 1464 . . . . . . . . . . . . . . 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 27488 . . . . . . . . . . . . . . . . . 18 ((𝑋 No 𝑤 No ) → (𝑋𝑤 ↔ (𝑋 <s 𝑤𝑤 <s 𝑋)))
4746ad2ant2lr 744 . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . . . . 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 855 . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . 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 455 . . . . . . . . . 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 453 . . . . . . . 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 679 . . . . . . 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 3026 . . . . . 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 455 . . . . 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 3144 . . . 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 27509 . . . . . 6 bday Fn No
77 ssrab2 4078 . . . . . 6 {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78 fnssintima 7363 . . . . . 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 688 . . . . 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 629 . . . . . 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 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 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 629 . . . . . 6 (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
91 simpr 483 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 No )
922, 4jca 510 . . . . . 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 7238 . . . . 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 1463 . . . 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 27602 . . . 4 ( L ‘𝑋) <<s ( R ‘𝑋)
100 eqscut 27541 . . . 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 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 480 . 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 1340 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 394  wo 843  w3a 1085   = wceq 1539  wcel 2104  wne 2938  wral 3059  {crab 3430  wss 3949  {csn 4629   cint 4951   class class class wbr 5149  cima 5680  Oncon0 6365   Fn wfn 6539  cfv 6544  (class class class)co 7413   No csur 27377   <s cslt 27378   bday cbday 27379   <<s csslt 27516   |s cscut 27518   M cmade 27572   O cold 27573   L cleft 27575   R cright 27576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-1o 8470  df-2o 8471  df-no 27380  df-slt 27381  df-bday 27382  df-sslt 27517  df-scut 27519  df-made 27577  df-old 27578  df-left 27580  df-right 27581
This theorem is referenced by:  madebday  27629  lrcut  27632
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