Theorem madebdaylemlrcut 28290

Description: Lemma for madebday 28291. If the inductive hypothesis of madebday 28291 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 28295 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 28251	. . 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 28252	. . 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 6885	. . . . . . . . 9 ⊢ (𝑋 = 𝑤 → (bday‘𝑋) = (bday‘𝑤))
6		eqimss 3966	. . . . . . . . 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 28158	. . . . . . . . . 10 ⊢ ((L‘𝑋) <<s {𝑤} → ∀𝑥 ∈ (L‘𝑋)∀𝑦 ∈ {𝑤}𝑥 <s 𝑦)
10		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
11		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤))
12	10, 11	ralsn 4638	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑤}𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤)
13	12	ralbii 3108	. . . . . . . . . 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 28158	. . . . . . . . . 10 ⊢ ({𝑤} <<s (R‘𝑋) → ∀𝑦 ∈ {𝑤}∀𝑥 ∈ (R‘𝑋)𝑦 <s 𝑥)
16		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥))
17	16	ralbidv 3185	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ (R‘𝑋)𝑦 <s 𝑥 ↔ ∀𝑥 ∈ (R‘𝑋)𝑤 <s 𝑥))
18	10, 17	ralsn 4638	. . . . . . . . . 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 28240	. . . . . . . . . . . . . . 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 3319	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (∀𝑥 ∈ (L‘𝑋)𝑥 <s 𝑤 ↔ ∀𝑥 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤))
24		rightval 28241	. . . . . . . . . . . . . . 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 3319	. . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋))
29	28	ralrab 3651	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {𝑧 ∈ (O‘(bday‘𝑋)) ∣ 𝑧 <s 𝑋}𝑥 <s 𝑤 ↔ ∀𝑥 ∈ (O‘(bday‘𝑋))(𝑥 <s 𝑋 → 𝑥 <s 𝑤))
30		breq2 5106	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥))
31	30	ralrab 3651	. . . . . . . . . . . . 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 28108	. . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . . . . 16 ⊢ (bday‘𝑋) ∈ On
39		bdayon 28143	. . . . . . . . . . . . . . . 16 ⊢ (bday‘𝑤) ∈ On
40		ontri1 6397	. . . . . . . . . . . . . . . 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 28289	. . . . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . . . 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 28113	. . . . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤))
50		breq2 5106	. . . . . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → (𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋))
57		breq1 5105	. . . . . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . . . . 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 3041	. . . . . 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 3154	. . . 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 28139	. . . . . 6 ⊢ bday Fn No
77		ssrab2 4005	. . . . . 6 ⊢ {𝑧 ∈ No ∣ ((L‘𝑋) <<s {𝑧} ∧ {𝑧} <<s (R‘𝑋))} ⊆ No
78		fnssintima 7372	. . . . . 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 4590	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
81	80	breq2d 5114	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((L‘𝑋) <<s {𝑧} ↔ (L‘𝑋) <<s {𝑤}))
82	80	breq1d 5112	. . . . . . 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 3651	. . . . 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 4590	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → {𝑧} = {𝑋})
88	87	breq2d 5114	. . . . . . 7 ⊢ (𝑧 = 𝑋 → ((L‘𝑋) <<s {𝑧} ↔ (L‘𝑋) <<s {𝑋}))
89	87	breq1d 5112	. . . . . . 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 3646	. . . . 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 7239	. . . . 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 1494	. . . 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 4922	. . . 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 3925	. 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 28253	. . . 4 ⊢ (L‘𝑋) <<s (R‘𝑋)
100		eqcuts 28176	. . . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412   ⊆ wss 3895  {csn 4584  ∩ cint 4906   class class class wbr 5102   “ cima 5653  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <s clts 28003  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150  Mcmade 28213  Ocold 28214  Lcleft 28216  Rcright 28217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-left 28221  df-right 28222

This theorem is used by:  madebday  28291  lrcut  28295