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Theorem madebday 33718
Description: A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
Assertion
Ref Expression
madebday ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday 𝑋) ⊆ 𝐴))

Proof of Theorem madebday
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madebdayim 33708 . . . 4 ((𝐴 ∈ On ∧ 𝑋 ∈ ( M ‘𝐴)) → ( bday 𝑋) ⊆ 𝐴)
21ex 416 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
32adantr 484 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
4 sseq2 3903 . . . . . . 7 (𝑎 = 𝑏 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝑏))
5 fveq2 6674 . . . . . . . 8 (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
65eleq2d 2818 . . . . . . 7 (𝑎 = 𝑏 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝑏)))
74, 6imbi12d 348 . . . . . 6 (𝑎 = 𝑏 → ((( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)) ↔ (( bday 𝑥) ⊆ 𝑏𝑥 ∈ ( M ‘𝑏))))
87ralbidv 3109 . . . . 5 (𝑎 = 𝑏 → (∀𝑥 No (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 No (( bday 𝑥) ⊆ 𝑏𝑥 ∈ ( M ‘𝑏))))
9 fveq2 6674 . . . . . . . 8 (𝑥 = 𝑦 → ( bday 𝑥) = ( bday 𝑦))
109sseq1d 3908 . . . . . . 7 (𝑥 = 𝑦 → (( bday 𝑥) ⊆ 𝑏 ↔ ( bday 𝑦) ⊆ 𝑏))
11 eleq1 2820 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘𝑏)))
1210, 11imbi12d 348 . . . . . 6 (𝑥 = 𝑦 → ((( bday 𝑥) ⊆ 𝑏𝑥 ∈ ( M ‘𝑏)) ↔ (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))))
1312cbvralvw 3349 . . . . 5 (∀𝑥 No (( bday 𝑥) ⊆ 𝑏𝑥 ∈ ( M ‘𝑏)) ↔ ∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)))
148, 13bitrdi 290 . . . 4 (𝑎 = 𝑏 → (∀𝑥 No (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))))
15 sseq2 3903 . . . . . 6 (𝑎 = 𝐴 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝐴))
16 fveq2 6674 . . . . . . 7 (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
1716eleq2d 2818 . . . . . 6 (𝑎 = 𝐴 → (𝑥 ∈ ( M ‘𝑎) ↔ 𝑥 ∈ ( M ‘𝐴)))
1815, 17imbi12d 348 . . . . 5 (𝑎 = 𝐴 → ((( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)) ↔ (( bday 𝑥) ⊆ 𝐴𝑥 ∈ ( M ‘𝐴))))
1918ralbidv 3109 . . . 4 (𝑎 = 𝐴 → (∀𝑥 No (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)) ↔ ∀𝑥 No (( bday 𝑥) ⊆ 𝐴𝑥 ∈ ( M ‘𝐴))))
20 bdayelon 33612 . . . . . . . . 9 ( bday 𝑥) ∈ On
21 onsseleq 6213 . . . . . . . . 9 ((( bday 𝑥) ∈ On ∧ 𝑎 ∈ On) → (( bday 𝑥) ⊆ 𝑎 ↔ (( bday 𝑥) ∈ 𝑎 ∨ ( bday 𝑥) = 𝑎)))
2220, 21mpan 690 . . . . . . . 8 (𝑎 ∈ On → (( bday 𝑥) ⊆ 𝑎 ↔ (( bday 𝑥) ∈ 𝑎 ∨ ( bday 𝑥) = 𝑎)))
2322ad2antrr 726 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → (( bday 𝑥) ⊆ 𝑎 ↔ (( bday 𝑥) ∈ 𝑎 ∨ ( bday 𝑥) = 𝑎)))
24 simpll 767 . . . . . . . . . . 11 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → 𝑎 ∈ On)
25 onelss 6214 . . . . . . . . . . . . 13 (𝑎 ∈ On → (( bday 𝑥) ∈ 𝑎 → ( bday 𝑥) ⊆ 𝑎))
2625ad2antrr 726 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → (( bday 𝑥) ∈ 𝑎 → ( bday 𝑥) ⊆ 𝑎))
2726imp 410 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → ( bday 𝑥) ⊆ 𝑎)
28 madess 33697 . . . . . . . . . . 11 ((( bday 𝑥) ∈ On ∧ 𝑎 ∈ On ∧ ( bday 𝑥) ⊆ 𝑎) → ( M ‘( bday 𝑥)) ⊆ ( M ‘𝑎))
2920, 24, 27, 28mp3an2ani 1469 . . . . . . . . . 10 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → ( M ‘( bday 𝑥)) ⊆ ( M ‘𝑎))
30 ssid 3899 . . . . . . . . . . 11 ( bday 𝑥) ⊆ ( bday 𝑥)
31 simpr 488 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → ( bday 𝑥) ∈ 𝑎)
32 simplr 769 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → 𝑥 No )
3331, 32jca 515 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → (( bday 𝑥) ∈ 𝑎𝑥 No ))
34 simpllr 776 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)))
35 sseq2 3903 . . . . . . . . . . . . . 14 (𝑏 = ( bday 𝑥) → (( bday 𝑦) ⊆ 𝑏 ↔ ( bday 𝑦) ⊆ ( bday 𝑥)))
36 fveq2 6674 . . . . . . . . . . . . . . 15 (𝑏 = ( bday 𝑥) → ( M ‘𝑏) = ( M ‘( bday 𝑥)))
3736eleq2d 2818 . . . . . . . . . . . . . 14 (𝑏 = ( bday 𝑥) → (𝑦 ∈ ( M ‘𝑏) ↔ 𝑦 ∈ ( M ‘( bday 𝑥))))
3835, 37imbi12d 348 . . . . . . . . . . . . 13 (𝑏 = ( bday 𝑥) → ((( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ↔ (( bday 𝑦) ⊆ ( bday 𝑥) → 𝑦 ∈ ( M ‘( bday 𝑥)))))
39 fveq2 6674 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ( bday 𝑦) = ( bday 𝑥))
4039sseq1d 3908 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (( bday 𝑦) ⊆ ( bday 𝑥) ↔ ( bday 𝑥) ⊆ ( bday 𝑥)))
41 eleq1 2820 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦 ∈ ( M ‘( bday 𝑥)) ↔ 𝑥 ∈ ( M ‘( bday 𝑥))))
4240, 41imbi12d 348 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((( bday 𝑦) ⊆ ( bday 𝑥) → 𝑦 ∈ ( M ‘( bday 𝑥))) ↔ (( bday 𝑥) ⊆ ( bday 𝑥) → 𝑥 ∈ ( M ‘( bday 𝑥)))))
4338, 42rspc2v 3536 . . . . . . . . . . . 12 ((( bday 𝑥) ∈ 𝑎𝑥 No ) → (∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → (( bday 𝑥) ⊆ ( bday 𝑥) → 𝑥 ∈ ( M ‘( bday 𝑥)))))
4433, 34, 43sylc 65 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → (( bday 𝑥) ⊆ ( bday 𝑥) → 𝑥 ∈ ( M ‘( bday 𝑥))))
4530, 44mpi 20 . . . . . . . . . 10 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘( bday 𝑥)))
4629, 45sseldd 3878 . . . . . . . . 9 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) ∧ ( bday 𝑥) ∈ 𝑎) → 𝑥 ∈ ( M ‘𝑎))
4746ex 416 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → (( bday 𝑥) ∈ 𝑎𝑥 ∈ ( M ‘𝑎)))
48 madebdaylemlrcut 33717 . . . . . . . . . . . 12 ((∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
4920a1i 11 . . . . . . . . . . . . . 14 (𝑥 No → ( bday 𝑥) ∈ On)
50 lltropt 33693 . . . . . . . . . . . . . 14 (𝑥 No → ( L ‘𝑥) <<s ( R ‘𝑥))
51 leftssold 33699 . . . . . . . . . . . . . 14 (𝑥 No → ( L ‘𝑥) ⊆ ( O ‘( bday 𝑥)))
52 rightssold 33700 . . . . . . . . . . . . . 14 (𝑥 No → ( R ‘𝑥) ⊆ ( O ‘( bday 𝑥)))
53 madecut 33703 . . . . . . . . . . . . . 14 (((( bday 𝑥) ∈ On ∧ ( L ‘𝑥) <<s ( R ‘𝑥)) ∧ (( L ‘𝑥) ⊆ ( O ‘( bday 𝑥)) ∧ ( R ‘𝑥) ⊆ ( O ‘( bday 𝑥)))) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday 𝑥)))
5449, 50, 51, 52, 53syl22anc 838 . . . . . . . . . . . . 13 (𝑥 No → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday 𝑥)))
5554adantl 485 . . . . . . . . . . . 12 ((∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → (( L ‘𝑥) |s ( R ‘𝑥)) ∈ ( M ‘( bday 𝑥)))
5648, 55eqeltrrd 2834 . . . . . . . . . . 11 ((∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → 𝑥 ∈ ( M ‘( bday 𝑥)))
57 raleq 3310 . . . . . . . . . . . . 13 (( bday 𝑥) = 𝑎 → (∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ↔ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))))
5857anbi1d 633 . . . . . . . . . . . 12 (( bday 𝑥) = 𝑎 → ((∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) ↔ (∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No )))
59 fveq2 6674 . . . . . . . . . . . . 13 (( bday 𝑥) = 𝑎 → ( M ‘( bday 𝑥)) = ( M ‘𝑎))
6059eleq2d 2818 . . . . . . . . . . . 12 (( bday 𝑥) = 𝑎 → (𝑥 ∈ ( M ‘( bday 𝑥)) ↔ 𝑥 ∈ ( M ‘𝑎)))
6158, 60imbi12d 348 . . . . . . . . . . 11 (( bday 𝑥) = 𝑎 → (((∀𝑏 ∈ ( bday 𝑥)∀𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → 𝑥 ∈ ( M ‘( bday 𝑥))) ↔ ((∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → 𝑥 ∈ ( M ‘𝑎))))
6256, 61mpbii 236 . . . . . . . . . 10 (( bday 𝑥) = 𝑎 → ((∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → 𝑥 ∈ ( M ‘𝑎)))
6362com12 32 . . . . . . . . 9 ((∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) ∧ 𝑥 No ) → (( bday 𝑥) = 𝑎𝑥 ∈ ( M ‘𝑎)))
6463adantll 714 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → (( bday 𝑥) = 𝑎𝑥 ∈ ( M ‘𝑎)))
6547, 64jaod 858 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → ((( bday 𝑥) ∈ 𝑎 ∨ ( bday 𝑥) = 𝑎) → 𝑥 ∈ ( M ‘𝑎)))
6623, 65sylbid 243 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) ∧ 𝑥 No ) → (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)))
6766ralrimiva 3096 . . . . 5 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏))) → ∀𝑥 No (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎)))
6867ex 416 . . . 4 (𝑎 ∈ On → (∀𝑏𝑎𝑦 No (( bday 𝑦) ⊆ 𝑏𝑦 ∈ ( M ‘𝑏)) → ∀𝑥 No (( bday 𝑥) ⊆ 𝑎𝑥 ∈ ( M ‘𝑎))))
6914, 19, 68tfis3 7591 . . 3 (𝐴 ∈ On → ∀𝑥 No (( bday 𝑥) ⊆ 𝐴𝑥 ∈ ( M ‘𝐴)))
70 fveq2 6674 . . . . . 6 (𝑥 = 𝑋 → ( bday 𝑥) = ( bday 𝑋))
7170sseq1d 3908 . . . . 5 (𝑥 = 𝑋 → (( bday 𝑥) ⊆ 𝐴 ↔ ( bday 𝑋) ⊆ 𝐴))
72 eleq1 2820 . . . . 5 (𝑥 = 𝑋 → (𝑥 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ ( M ‘𝐴)))
7371, 72imbi12d 348 . . . 4 (𝑥 = 𝑋 → ((( bday 𝑥) ⊆ 𝐴𝑥 ∈ ( M ‘𝐴)) ↔ (( bday 𝑋) ⊆ 𝐴𝑋 ∈ ( M ‘𝐴))))
7473rspccva 3525 . . 3 ((∀𝑥 No (( bday 𝑥) ⊆ 𝐴𝑥 ∈ ( M ‘𝐴)) ∧ 𝑋 No ) → (( bday 𝑋) ⊆ 𝐴𝑋 ∈ ( M ‘𝐴)))
7569, 74sylan 583 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (( bday 𝑋) ⊆ 𝐴𝑋 ∈ ( M ‘𝐴)))
763, 75impbid 215 1 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday 𝑋) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  wral 3053  wss 3843   class class class wbr 5030  Oncon0 6172  cfv 6339  (class class class)co 7170   No csur 33484   bday cbday 33486   <<s csslt 33616   |s cscut 33618   M cmade 33667   O cold 33668   L cleft 33670   R cright 33671
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
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 2075  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 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-wrecs 7976  df-recs 8037  df-1o 8131  df-2o 8132  df-no 33487  df-slt 33488  df-bday 33489  df-sslt 33617  df-scut 33619  df-made 33672  df-old 33673  df-left 33675  df-right 33676
This theorem is referenced by:  oldbday  33719  newbday  33720  lrcut  33721
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