MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  madebdaylemlrcut Structured version   Visualization version   GIF version

Theorem madebdaylemlrcut 27250
Description: Lemma for madebday 27251. If the inductive hypothesis of madebday 27251 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27254 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 27222 . . 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 27223 . . 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 6843 . . . . . . . . 9 (𝑋 = 𝑤 → ( bday 𝑋) = ( bday 𝑤))
6 eqimss 4001 . . . . . . . . 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 27152 . . . . . . . . . 10 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10 vex 3448 . . . . . . . . . . . 12 𝑤 ∈ V
11 breq2 5110 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑥 <s 𝑦𝑥 <s 𝑤))
1210, 11ralsn 4643 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦𝑥 <s 𝑤)
1312ralbii 3093 . . . . . . . . . 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 27152 . . . . . . . . . 10 ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16 breq1 5109 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦 <s 𝑥𝑤 <s 𝑥))
1716ralbidv 3171 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
1810, 17ralsn 4643 . . . . . . . . . 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 27215 . . . . . . . . . . . . . . 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 3312 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24 rightval 27216 . . . . . . . . . . . . . . 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 3312 . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 <s 𝑋𝑥 <s 𝑋))
2928ralrab 3652 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤))
30 breq2 5110 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑋 <s 𝑧𝑋 <s 𝑥))
3130ralrab 3652 . . . . . . . . . . . . 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 27110 . . . . . . . . . . . . . 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 27138 . . . . . . . . . . . . . . . 16 ( bday 𝑋) ∈ On
39 bdayelon 27138 . . . . . . . . . . . . . . . 16 ( bday 𝑤) ∈ On
40 ontri1 6352 . . . . . . . . . . . . . . . 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 27249 . . . . . . . . . . . . . . . 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 27115 . . . . . . . . . . . . . . . . . 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 5110 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑋 <s 𝑥𝑋 <s 𝑤))
50 breq2 5110 . . . . . . . . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑋𝑤 <s 𝑋))
57 breq1 5109 . . . . . . . . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . . 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 3028 . . . . . 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 3140 . . . 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 27135 . . . . . 6 bday Fn No
77 ssrab2 4038 . . . . . 6 {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78 fnssintima 7308 . . . . . 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 4597 . . . . . . . 8 (𝑧 = 𝑤 → {𝑧} = {𝑤})
8180breq2d 5118 . . . . . . 7 (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
8280breq1d 5116 . . . . . . 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 3652 . . . . 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 4597 . . . . . . . 8 (𝑧 = 𝑋 → {𝑧} = {𝑋})
8887breq2d 5118 . . . . . . 7 (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
8987breq1d 5116 . . . . . . 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 3648 . . . . 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 7184 . . . . 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 4925 . . . 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 3962 . 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 27224 . . . 4 ( L ‘𝑋) <<s ( R ‘𝑋)
100 eqscut 27166 . . . 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 2940  wral 3061  {crab 3406  wss 3911  {csn 4587   cint 4908   class class class wbr 5106  cima 5637  Oncon0 6318   Fn wfn 6492  cfv 6497  (class class class)co 7358   No csur 27004   <s cslt 27005   bday cbday 27006   <<s csslt 27142   |s cscut 27144   M cmade 27194   O cold 27195   L cleft 27197   R cright 27198
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145  df-made 27199  df-old 27200  df-left 27202  df-right 27203
This theorem is referenced by:  madebday  27251  lrcut  27254
  Copyright terms: Public domain W3C validator