Theorem madebdaylemlrcut 27937

Description: Lemma for madebday 27938. If the inductive hypothesis of madebday 27938 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27941 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		ssltleft 27909	. . 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		ssltright 27910	. . 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 6906	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → ( bday ‘𝑋) = ( bday ‘𝑤))
6		eqimss 4042	. . . . . . . . 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		ssltsep 27835	. . . . . . . . . 10 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5147	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4681	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤)
13	12	ralbii 3093	. . . . . . . . . 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		ssltsep 27835	. . . . . . . . . 10 ⊢ ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16		breq1 5146	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3178	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4681	. . . . . . . . . 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 613	. . . . . . . 8 ⊢ ((( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋)) → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
21		leftval 27902	. . . . . . . . . . . . . . 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 3326	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 27903	. . . . . . . . . . . . . . 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 3326	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
27	23, 26	anbi12d 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 5146	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3699	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5147	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3699	. . . . . . . . . . . . 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 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 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 727	. . . . . . . . . 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		sltirr 27791	. . . . . . . . . . . . . 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		bdayelon 27821	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑋) ∈ On
39		bdayelon 27821	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑤) ∈ On
40		ontri1 6418	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑤) ∈ On) → (( bday ‘𝑋) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))
41	38, 39, 40	mp2an 692	. . . . . . . . . . . . . . 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 27936	. . . . . . . . . . . . . . . 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 1468	. . . . . . . . . . . . . . 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		slttrine 27796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ No ∧ 𝑤 ∈ No ) → (𝑋 ≠ 𝑤 ↔ (𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋)))
47	46	ad2ant2lr 748	. . . . . . . . . . . . . . . . 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 5147	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5147	. . . . . . . . . . . . . . . . . . . . . 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 3619	. . . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5146	. . . . . . . . . . . . . . . . . . . . . 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 3619	. . . . . . . . . . . . . . . . . . . 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 683	. . . . . . 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 3028	. . . . . 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 3146	. . . 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 27818	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4080	. . . . . 6 ⊢ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78		fnssintima 7382	. . . . . 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 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))
80		sneq 4636	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5155	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
82	80	breq1d 5153	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
83	81, 82	anbi12d 632	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
84	83	ralrab 3699	. . . . 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 4636	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5155	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
89	87	breq1d 5153	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
90	88, 89	anbi12d 632	. . . . . 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 3694	. . . . 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 7253	. . . . 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 1467	. . . 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 4963	. . . 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 4001	. 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		lltropt 27911	. . . 4 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
100		eqscut 27850	. . . 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 690	. . 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 1343	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 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  {crab 3436   ⊆ wss 3951  {csn 4626  ∩ cint 4946   class class class wbr 5143   “ cima 5688  Oncon0 6384   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   |s cscut 27827   M cmade 27881   O cold 27882   L cleft 27884   R cright 27885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-left 27889  df-right 27890

This theorem is referenced by:  madebday  27938  lrcut  27941