Theorem madebdaylemlrcut 27891

Description: Lemma for madebday 27892. If the inductive hypothesis of madebday 27892 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27896 holds. (Contributed by Scott Fenton, 19-Aug-2024.)

Assertion

Ref	Expression
madebdaylemlrcut	⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( 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.

Step	Hyp	Ref	Expression
1		sltsleft 27852	. . 3 ⊢ (𝑋 ∈ No → ( L ‘𝑋) <<s {𝑋})
2	1	adantl 481	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ( L ‘𝑋) <<s {𝑋})
3		sltsright 27853	. . 3 ⊢ (𝑋 ∈ No → {𝑋} <<s ( R ‘𝑋))
4	3	adantl 481	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → {𝑋} <<s ( R ‘𝑋))
5		fveq2 6840	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → ( bday ‘𝑋) = ( bday ‘𝑤))
6		eqimss 3980	. . . . . . . . 9 ⊢ (( bday ‘𝑋) = ( bday ‘𝑤) → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑋 = 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
8	7	a1i 11	. . . . . . 7 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → (𝑋 = 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
9		sltssep 27759	. . . . . . . . . 10 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3433	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5089	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4625	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤)
13	12	ralbii 3083	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
14	9, 13	sylib 218	. . . . . . . . 9 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
15		sltssep 27759	. . . . . . . . . 10 ⊢ ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16		breq1 5088	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3160	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4625	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
19	15, 18	sylib 218	. . . . . . . . 9 ⊢ ({𝑤} <<s ( R ‘𝑋) → ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
20	14, 19	anim12i 614	. . . . . . . 8 ⊢ ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21		leftval 27841	. . . . . . . . . . . . . . 15 ⊢ ( L ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ No → ( L ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋})
23	22	raleqdv 3295	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 27842	. . . . . . . . . . . . . . 15 ⊢ ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}
25	24	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ No → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧})
26	25	raleqdv 3295	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
27	23, 26	anbi12d 633	. . . . . . . . . . . 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 5088	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3640	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5089	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3640	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥 ↔ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥))
32	29, 31	anbi12i 629	. . . . . . . . . . . 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 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))
33	27, 32	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑋 ∈ No → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) ↔ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥))))
34	33	ad2antlr 728	. . . . . . . . . 10 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) ↔ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥))))
35		simplrl 777	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → 𝑤 ∈ No )
36		ltsirr 27710	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ No → ¬ 𝑤 <s 𝑤)
37	35, 36	syl 17	. . . . . . . . . . . . 13 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → ¬ 𝑤 <s 𝑤)
38		bdayon 27744	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑋) ∈ On
39		bdayon 27744	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑤) ∈ On
40		ontri1 6357	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑤) ∈ On) → (( bday ‘𝑋) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))
41	38, 39, 40	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))
42	41	con2bii 357	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ↔ ¬ ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
43		simplll 775	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → ∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)))
44		madebdaylemold 27890	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑤 ∈ No ) → (( bday ‘𝑤) ∈ ( bday ‘𝑋) → 𝑤 ∈ ( O ‘( bday ‘𝑋))))
45	38, 43, 35, 44	mp3an2i 1469	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → (( bday ‘𝑤) ∈ ( bday ‘𝑋) → 𝑤 ∈ ( O ‘( bday ‘𝑋))))
46		ltstrine 27715	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ No ∧ 𝑤 ∈ No ) → (𝑋 ≠ 𝑤 ↔ (𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋)))
47	46	ad2ant2lr 749	. . . . . . . . . . . . . . . . 17 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑋 ≠ 𝑤 ↔ (𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋)))
48		simprrr 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥))
49		breq2 5089	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5089	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑤 <s 𝑥 ↔ 𝑤 <s 𝑤))
51	49, 50	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((𝑋 <s 𝑥 → 𝑤 <s 𝑥) ↔ (𝑋 <s 𝑤 → 𝑤 <s 𝑤)))
52	51	rspccv 3561	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → (𝑋 <s 𝑤 → 𝑤 <s 𝑤)))
53	48, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → (𝑋 <s 𝑤 → 𝑤 <s 𝑤)))
54	53	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑋 <s 𝑤 → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → 𝑤 <s 𝑤)))
55		simprrl 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
56		breq1 5088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑤 ↔ 𝑤 <s 𝑤))
58	56, 57	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((𝑥 <s 𝑋 → 𝑥 <s 𝑤) ↔ (𝑤 <s 𝑋 → 𝑤 <s 𝑤)))
59	58	rspccv 3561	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → (𝑤 <s 𝑋 → 𝑤 <s 𝑤)))
60	55, 59	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → (𝑤 <s 𝑋 → 𝑤 <s 𝑤)))
61	60	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑤 <s 𝑋 → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → 𝑤 <s 𝑤)))
62	54, 61	jaod 860	. . . . . . . . . . . . . . . . 17 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → ((𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → 𝑤 <s 𝑤)))
63	47, 62	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑋 ≠ 𝑤 → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → 𝑤 <s 𝑤)))
64	63	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → (𝑤 ∈ ( O ‘( bday ‘𝑋)) → 𝑤 <s 𝑤))
65	45, 64	syld 47	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → (( bday ‘𝑤) ∈ ( bday ‘𝑋) → 𝑤 <s 𝑤))
66	42, 65	biimtrrid 243	. . . . . . . . . . . . 13 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → (¬ ( bday ‘𝑋) ⊆ ( bday ‘𝑤) → 𝑤 <s 𝑤))
67	37, 66	mt3d 148	. . . . . . . . . . . 12 ⊢ ((((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) ∧ 𝑋 ≠ 𝑤) → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
68	67	ex 412	. . . . . . . . . . 11 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)))) → (𝑋 ≠ 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
69	68	expr 456	. . . . . . . . . 10 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ((∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤) ∧ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑋 <s 𝑥 → 𝑤 <s 𝑥)) → (𝑋 ≠ 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))))
70	34, 69	sylbid 240	. . . . . . . . 9 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ((∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥) → (𝑋 ≠ 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))))
71	70	impr 454	. . . . . . . 8 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))) → (𝑋 ≠ 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
72	20, 71	sylanr2 684	. . . . . . 7 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → (𝑋 ≠ 𝑤 → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
73	8, 72	pm2.61dne 3018	. . . . . 6 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ (𝑤 ∈ No ∧ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)))) → ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
74	73	expr 456	. . . . 5 ⊢ (((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
75	74	ralrimiva 3129	. . . 4 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ∀𝑤 ∈ No ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → ( bday ‘𝑋) ⊆ ( bday ‘𝑤)))
76		bdayfn 27741	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4020	. . . . . 6 ⊢ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78		fnssintima 7317	. . . . . 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)))
79	76, 77, 78	mp2an 693	. . . . 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 4577	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5097	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
82	80	breq1d 5095	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
83	81, 82	anbi12d 633	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
84	83	ralrab 3640	. . . . 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)))
85	79, 84	bitri 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)))
86	75, 85	sylibr 234	. . 3 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ( bday ‘𝑋) ⊆ ∩ ( bday “ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
87		sneq 4577	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5097	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
89	87	breq1d 5095	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
90	88, 89	anbi12d 633	. . . . . 6 ⊢ (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
91		simpr 484	. . . . . 6 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → 𝑋 ∈ No )
92	2, 4	jca 511	. . . . . 6 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋)))
93	90, 91, 92	elrabd 3636	. . . . 5 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → 𝑋 ∈ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))})
94		fnfvima 7188	. . . . 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))}))
95	76, 77, 93, 94	mp3an12i 1468	. . . 4 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ( bday ‘𝑋) ∈ ( bday “ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
96		intss1 4905	. . . 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))
97	95, 96	syl 17	. . 3 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ∩ ( bday “ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}) ⊆ ( bday ‘𝑋))
98	86, 97	eqssd 3939	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ( bday ‘𝑋) = ∩ ( bday “ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))}))
99		lltr 27854	. . . 4 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
100		eqcuts 27777	. . . 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))}))))
101	99, 100	mpan 691	. . 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))}))))
102	101	adantl 481	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( 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))}))))
103	2, 4, 98, 102	mpbir3and 1344	1 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389   ⊆ wss 3889  {csn 4567  ∩ cint 4889   class class class wbr 5085   “ cima 5634  Oncon0 6323   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   bday cbday 27605   <<s cslts 27749   |s ccuts 27751   M cmade 27814   O cold 27815   L cleft 27817   R cright 27818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823

This theorem is referenced by:  madebday  27892  lrcut  27896