Theorem madebdaylemlrcut 27905

Description: Lemma for madebday 27906. If the inductive hypothesis of madebday 27906 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27910 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 27866	. . 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 27867	. . 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 6834	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → ( bday ‘𝑋) = ( bday ‘𝑤))
6		eqimss 3981	. . . . . . . . 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 27773	. . . . . . . . . 10 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3434	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5090	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4626	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤)
13	12	ralbii 3084	. . . . . . . . . 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 27773	. . . . . . . . . 10 ⊢ ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16		breq1 5089	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3161	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4626	. . . . . . . . . 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 27855	. . . . . . . . . . . . . . 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 3296	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 27856	. . . . . . . . . . . . . . 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 3296	. . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3641	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5090	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3641	. . . . . . . . . . . . 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 27724	. . . . . . . . . . . . . 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 27758	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑋) ∈ On
39		bdayon 27758	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑤) ∈ On
40		ontri1 6351	. . . . . . . . . . . . . . . 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 27904	. . . . . . . . . . . . . . . 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 27729	. . . . . . . . . . . . . . . . . 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 5090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5090	. . . . . . . . . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5089	. . . . . . . . . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . . . . . . . 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 3019	. . . . . 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 3130	. . . 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 27755	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4021	. . . . . 6 ⊢ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78		fnssintima 7310	. . . . . 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 4578	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5098	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
82	80	breq1d 5096	. . . . . . 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 3641	. . . . 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 4578	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5098	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
89	87	breq1d 5096	. . . . . . 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 3637	. . . . 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 7181	. . . . 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 4906	. . . 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 3940	. 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 27868	. . . 4 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
100		eqcuts 27791	. . . 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 2933  ∀wral 3052  {crab 3390   ⊆ wss 3890  {csn 4568  ∩ cint 4890   class class class wbr 5086   “ cima 5627  Oncon0 6317   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   No csur 27617   <s clts 27618   bday cbday 27619   <<s cslts 27763   |s ccuts 27765   M cmade 27828   O cold 27829   L cleft 27831   R cright 27832

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-made 27833  df-old 27834  df-left 27836  df-right 27837

This theorem is referenced by:  madebday  27906  lrcut  27910