Theorem madebdaylemlrcut 28318

Description: Lemma for madebday 28319. If the inductive hypothesis of madebday 28319 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 28323 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 28279	. . 3 ⊢ (𝑋 ∈ No → (L‘𝑋) <<s {𝑋})
2	1	adantl 487	. 2 ⊢ ((∀𝑏 ∈ (bday‘𝑋)∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → (L‘𝑋) <<s {𝑋})
3		sltsright 28280	. . 3 ⊢ (𝑋 ∈ No → {𝑋} <<s (R‘𝑋))
4	3	adantl 487	. 2 ⊢ ((∀𝑏 ∈ (bday‘𝑋)∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → {𝑋} <<s (R‘𝑋))
5		fveq2 6911	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → (bday‘𝑋) = (bday‘𝑤))
6		eqimss 3971	. . . . . . . . 9 ⊢ ((bday‘𝑋) = (bday‘𝑤) → (bday‘𝑋) ⊆ (bday‘𝑤))
7	5, 6	syl 18	. . . . . . . 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 28186	. . . . . . . . . 10 ⊢ ((L‘𝑋) <<s {𝑤} → ∀𝑥 ∈ (L‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5110	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4642	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤)
13	12	ralbii 3112	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (L‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ ∀𝑥 ∈ (L‘𝑋)𝑥 <s 𝑤)
14	9, 13	sylib 221	. . . . . . . . 9 ⊢ ((L‘𝑋) <<s {𝑤} → ∀𝑥 ∈ (L‘𝑋)𝑥 <s 𝑤)
15		sltssep 28186	. . . . . . . . . 10 ⊢ ({𝑤} <<s (R‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ (R‘𝑋)𝑦 <s 𝑥)
16		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3189	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ (R‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4642	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑤}∀𝑥 ∈ (R‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥)
19	15, 18	sylib 221	. . . . . . . . 9 ⊢ ({𝑤} <<s (R‘𝑋) → ∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥)
20	14, 19	anim12i 625	. . . . . . . 8 ⊢ (((L‘𝑋) <<s {𝑤} ∧ {𝑤} <<s (R‘𝑋)) → (∀𝑥 ∈ (L‘𝑋)𝑥 <s 𝑤 ∧ ∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥))
21		leftval 28268	. . . . . . . . . . . . . . 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 3323	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ (L‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 28269	. . . . . . . . . . . . . . 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 3323	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥 ↔ ∀𝑥 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑋 <s 𝑧}𝑤 <s 𝑥))
27	23, 26	anbi12d 644	. . . . . . . . . . . 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 5109	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3655	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ (O‘(bday‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5110	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3655	. . . . . . . . . . . . 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 640	. . . . . . . . . . . 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 290	. . . . . . . . . . 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 740	. . . . . . . . . 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 789	. . . . . . . . . . . . . 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 28136	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ No → ¬ 𝑤 <s 𝑤)
37	35, 36	syl 18	. . . . . . . . . . . . 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 28171	. . . . . . . . . . . . . . . 16 ⊢ (bday‘𝑋) ∈ On
39		bdayon 28171	. . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑋) ⊆ (bday‘𝑤) ↔ ¬ (bday‘𝑤) ∈ (bday‘𝑋))
42	41	con2bii 360	. . . . . . . . . . . . . 14 ⊢ ((bday‘𝑤) ∈ (bday‘𝑋) ↔ ¬ (bday‘𝑋) ⊆ (bday‘𝑤))
43		simplll 787	. . . . . . . . . . . . . . . 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 28317	. . . . . . . . . . . . . . . 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 1496	. . . . . . . . . . . . . . 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 28141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ No ∧ 𝑤 ∈ No) → (𝑋 ≠ 𝑤 ↔ (𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋)))
47	46	ad2ant2lr 761	. . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑤 <s 𝑥 ↔ 𝑤 <s 𝑤))
51	49, 50	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((𝑋 <s 𝑥 → 𝑤 <s 𝑥) ↔ (𝑋 <s 𝑤 → 𝑤 <s 𝑤)))
52	51	rspccv 3577	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5109	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑤 ↔ 𝑤 <s 𝑤))
58	56, 57	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → ((𝑥 <s 𝑋 → 𝑥 <s 𝑤) ↔ (𝑤 <s 𝑋 → 𝑤 <s 𝑤)))
59	58	rspccv 3577	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . 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 873	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . 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 246	. . . . . . . . . . . . 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 149	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 462	. . . . . . . . . 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 243	. . . . . . . . 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 460	. . . . . . . 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 696	. . . . . . 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 3045	. . . . . 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 462	. . . . 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 3158	. . . 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 28167	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4008	. . . . . 6 ⊢ {𝑧 ∈ No ∣ ((L‘𝑋) <<s {𝑧} ∧ {𝑧} <<s (R‘𝑋))} ⊆ No
78		fnssintima 7398	. . . . . 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 705	. . . . 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 4594	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5118	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((L‘𝑋) <<s {𝑧} ↔ (L‘𝑋) <<s {𝑤}))
82	80	breq1d 5116	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ({𝑧} <<s (R‘𝑋) ↔ {𝑤} <<s (R‘𝑋)))
83	81, 82	anbi12d 644	. . . . . 6 ⊢ (𝑧 = 𝑤 → (((L‘𝑋) <<s {𝑧} ∧ {𝑧} <<s (R‘𝑋)) ↔ ((L‘𝑋) <<s {𝑤} ∧ {𝑤} <<s (R‘𝑋))))
84	83	ralrab 3655	. . . . 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 278	. . . 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 237	. . 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 4594	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5118	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ((L‘𝑋) <<s {𝑧} ↔ (L‘𝑋) <<s {𝑋}))
89	87	breq1d 5116	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ({𝑧} <<s (R‘𝑋) ↔ {𝑋} <<s (R‘𝑋)))
90	88, 89	anbi12d 644	. . . . . 6 ⊢ (𝑧 = 𝑋 → (((L‘𝑋) <<s {𝑧} ∧ {𝑧} <<s (R‘𝑋)) ↔ ((L‘𝑋) <<s {𝑋} ∧ {𝑋} <<s (R‘𝑋))))
91		simpr 490	. . . . . 6 ⊢ ((∀𝑏 ∈ (bday‘𝑋)∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → 𝑋 ∈ No)
92	2, 4	jca 521	. . . . . 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 3650	. . . . 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 7265	. . . . 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 1495	. . . 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 4926	. . . 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 18	. . 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 3930	. 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 28281	. . . 4 ⊢ (L‘𝑋) <<s (R‘𝑋)
100		eqcuts 28204	. . . 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 703	. . 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 487	. 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 1361	1 ⊢ ((∀𝑏 ∈ (bday‘𝑋)∀𝑦 ∈ No ((bday‘𝑦) ⊆ 𝑏 → 𝑦 ∈ (M‘𝑏)) ∧ 𝑋 ∈ No) → ((L‘𝑋) |s (R‘𝑋)) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416   ⊆ wss 3900  {csn 4588  ∩ cint 4910   class class class wbr 5106   “ cima 5659  Oncon0 6383   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <s clts 28031  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178  Mcmade 28241  Ocold 28242  Lcleft 28244  Rcright 28245

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246  df-old 28247  df-left 28249  df-right 28250

This theorem is used by:  madebday  28319  lrcut  28323