Theorem madebdaylemlrcut 27382

Description: Lemma for madebday 27383. If the inductive hypothesis of madebday 27383 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 27386 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 27354	. . 3 ⊢ (𝑋 ∈ No → ( L ‘𝑋) <<s {𝑋})
2	1	adantl 482	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → ( L ‘𝑋) <<s {𝑋})
3		ssltright 27355	. . 3 ⊢ (𝑋 ∈ No → {𝑋} <<s ( R ‘𝑋))
4	3	adantl 482	. 2 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → {𝑋} <<s ( R ‘𝑋))
5		fveq2 6888	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → ( bday ‘𝑋) = ( bday ‘𝑤))
6		eqimss 4039	. . . . . . . . 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 27281	. . . . . . . . . 10 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5151	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4684	. . . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (( L ‘𝑋) <<s {𝑤} → ∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤)
15		ssltsep 27281	. . . . . . . . . 10 ⊢ ({𝑤} <<s ( R ‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥)
16		breq1 5150	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3177	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4684	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑤}∀𝑥 ∈ ( R ‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥)
19	15, 18	sylib 217	. . . . . . . . 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 27347	. . . . . . . . . . . . . . 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 3325	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( L ‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 27348	. . . . . . . . . . . . . . 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 3325	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ ( R ‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
27	23, 26	anbi12d 631	. . . . . . . . . . . 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 5150	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3688	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ ( O ‘( bday ‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5151	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3688	. . . . . . . . . . . . 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 627	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 725	. . . . . . . . . 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 775	. . . . . . . . . . . . . 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 27238	. . . . . . . . . . . . . 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 27267	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑋) ∈ On
39		bdayelon 27267	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘𝑤) ∈ On
40		ontri1 6395	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑤) ∈ On) → (( bday ‘𝑋) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋)))
41	38, 39, 40	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ ( bday ‘𝑋))
42	41	con2bii 357	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑤) ∈ ( bday ‘𝑋) ↔ ¬ ( bday ‘𝑋) ⊆ ( bday ‘𝑤))
43		simplll 773	. . . . . . . . . . . . . . . 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 27381	. . . . . . . . . . . . . . . 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 1466	. . . . . . . . . . . . . . 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 27243	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ No ∧ 𝑤 ∈ No ) → (𝑋 ≠ 𝑤 ↔ (𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋)))
47	46	ad2ant2lr 746	. . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5151	. . . . . . . . . . . . . . . . . . . . . 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 3609	. . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . 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 5150	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5150	. . . . . . . . . . . . . . . . . . . . . 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 3609	. . . . . . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 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 413	. . . . . . . . . . 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 457	. . . . . . . . . 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 239	. . . . . . . . 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 455	. . . . . . . 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 681	. . . . . . 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 457	. . . . 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 27264	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4076	. . . . . 6 ⊢ {𝑧 ∈ No ∣ (( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋))} ⊆ No
78		fnssintima 7355	. . . . . 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 690	. . . . 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 4637	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5159	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑤}))
82	80	breq1d 5157	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑤} <<s ( R ‘𝑋)))
83	81, 82	anbi12d 631	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑤} ∧ {𝑤} <<s ( R ‘𝑋))))
84	83	ralrab 3688	. . . . 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 274	. . . 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 233	. . 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 4637	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5159	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (( L ‘𝑋) <<s {𝑧} ↔ ( L ‘𝑋) <<s {𝑋}))
89	87	breq1d 5157	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ({𝑧} <<s ( R ‘𝑋) ↔ {𝑋} <<s ( R ‘𝑋)))
90	88, 89	anbi12d 631	. . . . . 6 ⊢ (𝑧 = 𝑋 → ((( L ‘𝑋) <<s {𝑧} ∧ {𝑧} <<s ( R ‘𝑋)) ↔ (( L ‘𝑋) <<s {𝑋} ∧ {𝑋} <<s ( R ‘𝑋))))
91		simpr 485	. . . . . 6 ⊢ ((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → 𝑋 ∈ No )
92	2, 4	jca 512	. . . . . 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 3684	. . . . 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 7231	. . . . 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 1465	. . . 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 4966	. . . 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 3998	. 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 27356	. . . 4 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
100		eqscut 27295	. . . 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 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))}))))
102	101	adantl 482	. 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 1342	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 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  {crab 3432   ⊆ wss 3947  {csn 4627  ∩ cint 4949   class class class wbr 5147   “ cima 5678  Oncon0 6361   Fn wfn 6535  ‘cfv 6540  (class class class)co 7405   No csur 27132   <s cslt 27133   bday cbday 27134   <<s csslt 27271   |s cscut 27273   M cmade 27326   O cold 27327   L cleft 27329   R cright 27330

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-made 27331  df-old 27332  df-left 27334  df-right 27335

This theorem is referenced by:  madebday  27383  lrcut  27386