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Theorem madebdaylemlrcut 33709
Description: Lemma for madebday 33710. If the inductive hypothesis of madebday 33710 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 33713 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 33683 . . 3 (𝑋 No → ( L ‘𝑋) <<s {𝑋})
21adantl 485 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → ( L ‘𝑋) <<s {𝑋})
3 ssltright 33684 . . 3 (𝑋 No → {𝑋} <<s ( R ‘𝑋))
43adantl 485 . 2 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → {𝑋} <<s ( R ‘𝑋))
5 fveq2 6668 . . . . . . . . 9 (𝑋 = 𝑤 → ( bday 𝑋) = ( bday 𝑤))
6 eqimss 3931 . . . . . . . . 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 33618 . . . . . . . . . 10 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10 vex 3401 . . . . . . . . . . . 12 𝑤 ∈ V
11 breq2 5031 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑥 <s 𝑦𝑥 <s 𝑤))
1210, 11ralsn 4569 . . . . . . . . . . 11 (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦𝑥 <s 𝑤)
1312ralbii 3080 . . . . . . . . . 10 (∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
149, 13sylib 221 . . . . . . . . 9 (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
15 ssltsep 33618 . . . . . . . . . 10 ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16 breq1 5030 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑦 <s 𝑥𝑤 <s 𝑥))
1716ralbidv 3109 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
1810, 17ralsn 4569 . . . . . . . . . 10 (∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
1915, 18sylib 221 . . . . . . . . 9 ({𝑤} <<s ( R ‘𝑋) → ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
2014, 19anim12i 616 . . . . . . . 8 ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21 leftval 33676 . . . . . . . . . . . . . 14 (𝑋 No → ( L ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋})
2221raleqdv 3315 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
23 rightval 33677 . . . . . . . . . . . . . 14 (𝑋 No → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧})
2423raleqdv 3315 . . . . . . . . . . . . 13 (𝑋 No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
2522, 24anbi12d 634 . . . . . . . . . . . 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)))
26 breq1 5030 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 <s 𝑋𝑥 <s 𝑋))
2726ralrab 3591 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤))
28 breq2 5031 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑋 <s 𝑧𝑋 <s 𝑥))
2928ralrab 3591 . . . . . . . . . . . . 13 (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥 ↔ ∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥))
3027, 29anbi12i 630 . . . . . . . . . . . 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)))
3125, 30bitrdi 290 . . . . . . . . . . 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))))
3231ad2antlr 727 . . . . . . . . . 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))))
33 simplrl 777 . . . . . . . . . . . . . 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 )
34 sltirr 33582 . . . . . . . . . . . . . 14 (𝑤 No → ¬ 𝑤 <s 𝑤)
3533, 34syl 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)
36 bdayelon 33604 . . . . . . . . . . . . . . . 16 ( bday 𝑋) ∈ On
37 bdayelon 33604 . . . . . . . . . . . . . . . 16 ( bday 𝑤) ∈ On
38 ontri1 6200 . . . . . . . . . . . . . . . 16 ((( bday 𝑋) ∈ On ∧ ( bday 𝑤) ∈ On) → (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋)))
3936, 37, 38mp2an 692 . . . . . . . . . . . . . . 15 (( bday 𝑋) ⊆ ( bday 𝑤) ↔ ¬ ( bday 𝑤) ∈ ( bday 𝑋))
4039con2bii 361 . . . . . . . . . . . . . 14 (( bday 𝑤) ∈ ( bday 𝑋) ↔ ¬ ( bday 𝑋) ⊆ ( bday 𝑤))
41 simplll 775 . . . . . . . . . . . . . . . 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)))
42 madebdaylemold 33708 . . . . . . . . . . . . . . . 16 ((( bday 𝑋) ∈ On ∧ ∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑤 No ) → (( bday 𝑤) ∈ ( bday 𝑋) → 𝑤 ∈ ( O ‘( bday 𝑋))))
4336, 41, 33, 42mp3an2i 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))))
44 slttrine 33587 . . . . . . . . . . . . . . . . . 18 ((𝑋 No 𝑤 No ) → (𝑋𝑤 ↔ (𝑋 <s 𝑤𝑤 <s 𝑋)))
4544ad2ant2lr 748 . . . . . . . . . . . . . . . . 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)))
46 simprrr 782 . . . . . . . . . . . . . . . . . . . 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))
47 breq2 5031 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑋 <s 𝑥𝑋 <s 𝑤))
48 breq2 5031 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑤 <s 𝑥𝑤 <s 𝑤))
4947, 48imbi12d 348 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑋 <s 𝑥𝑤 <s 𝑥) ↔ (𝑋 <s 𝑤𝑤 <s 𝑤)))
5049rspccv 3521 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑋 <s 𝑥𝑤 <s 𝑥) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑋 <s 𝑤𝑤 <s 𝑤)))
5146, 50syl 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)))
5251com23 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)))
53 simprrl 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 𝑋𝑥 <s 𝑤))
54 breq1 5030 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑋𝑤 <s 𝑋))
55 breq1 5030 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → (𝑥 <s 𝑤𝑤 <s 𝑤))
5654, 55imbi12d 348 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → ((𝑥 <s 𝑋𝑥 <s 𝑤) ↔ (𝑤 <s 𝑋𝑤 <s 𝑤)))
5756rspccv 3521 . . . . . . . . . . . . . . . . . . . 20 (∀𝑥 ∈ ( O ‘( bday 𝑋))(𝑥 <s 𝑋𝑥 <s 𝑤) → (𝑤 ∈ ( O ‘( bday 𝑋)) → (𝑤 <s 𝑋𝑤 <s 𝑤)))
5853, 57syl 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)))
5958com23 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)))
6052, 59jaod 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)))
6145, 60sylbid 243 . . . . . . . . . . . . . . . 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)))
6261imp 410 . . . . . . . . . . . . . . 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))
6343, 62syld 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))
6440, 63syl5bir 246 . . . . . . . . . . . . 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))
6535, 64mt3d 150 . . . . . . . . . . . 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))
6665ex 416 . . . . . . . . . . 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)))
6766expr 460 . . . . . . . . . 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))))
6832, 67sylbid 243 . . . . . . . . 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))))
6968impr 458 . . . . . . . 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)))
7020, 69sylanr2 683 . . . . . . 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)))
718, 70pm2.61dne 3020 . . . . . 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))
7271expr 460 . . . . 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)))
7372ralrimiva 3096 . . . 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)))
74 bdayfn 33601 . . . . . 6 bday Fn No
75 ssrab2 3967 . . . . . 6 {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
76 fnssintima 33241 . . . . . 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)))
7774, 75, 76mp2an 692 . . . . 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))
78 sneq 4523 . . . . . . . 8 (𝑧 = 𝑤 → {𝑧} = {𝑤})
7978breq2d 5039 . . . . . . 7 (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
8078breq1d 5037 . . . . . . 7 (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
8179, 80anbi12d 634 . . . . . 6 (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
8281ralrab 3591 . . . . 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)))
8377, 82bitri 278 . . . 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)))
8473, 83sylibr 237 . . 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))}))
85 sneq 4523 . . . . . . . 8 (𝑧 = 𝑋 → {𝑧} = {𝑋})
8685breq2d 5039 . . . . . . 7 (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
8785breq1d 5037 . . . . . . 7 (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
8886, 87anbi12d 634 . . . . . 6 (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
89 simpr 488 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 No )
902, 4jca 515 . . . . . 6 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋)))
9188, 89, 90elrabd 3587 . . . . 5 ((∀𝑏 ∈ ( bday 𝑋)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 No ) → 𝑋 ∈ {𝑧 No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))})
92 fnfvima 7000 . . . . 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))}))
9374, 75, 91, 92mp3an12i 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))}))
94 intss1 4848 . . . 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))
9593, 94syl 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))
9684, 95eqssd 3892 . 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))}))
97 lltropt 33685 . . . 4 (𝑋 No → ( L ‘𝑋) <<s ( R ‘𝑋))
98 eqscut 33632 . . . 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))}))))
9997, 98mpancom 688 . . 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))}))))
10099adantl 485 . 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))}))))
1012, 4, 96, 100mpbir3and 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 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2113  wne 2934  wral 3053  {crab 3057  wss 3841  {csn 4513   cint 4833   class class class wbr 5027  cima 5522  Oncon0 6166   Fn wfn 6328  cfv 6333  (class class class)co 7164   No csur 33476   <s cslt 33477   bday cbday 33478   <<s csslt 33608   |s cscut 33610   M cmade 33659   O cold 33660   L cleft 33662   R cright 33663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-wrecs 7969  df-recs 8030  df-1o 8124  df-2o 8125  df-no 33479  df-slt 33480  df-bday 33481  df-sslt 33609  df-scut 33611  df-made 33664  df-old 33665  df-left 33667  df-right 33668
This theorem is referenced by:  madebday  33710  lrcut  33713
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