Theorem eqcuts2 28205

Description: Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
eqcuts2	⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))))

Distinct variable groups:   𝑦,𝐿   𝑦,𝑅   𝑦,𝑋

Proof of Theorem eqcuts2

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcuts 28204	. 2 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ (bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))))
2		eqss 3928	. . . . 5 ⊢ ((bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ (bday‘𝑋)))
3		sneq 4594	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	breq2d 5118	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
5	3	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
6	4, 5	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
7	6	elrab3 3649	. . . . . . . . . 10 ⊢ (𝑋 ∈ No → (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
8	7	adantl 487	. . . . . . . . 9 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
9	8	biimpar 483	. . . . . . . 8 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → 𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
10		bdayfn 28167	. . . . . . . . 9 ⊢ bday Fn No
11		ssrab2 4008	. . . . . . . . 9 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12		fnfvima 7265	. . . . . . . . 9 ⊢ ((bday Fn No ∧ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No ∧ 𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → (bday‘𝑋) ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
13	10, 11, 12	mp3an12 1481	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → (bday‘𝑋) ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14		intss1 4926	. . . . . . . 8 ⊢ ((bday‘𝑋) ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ (bday‘𝑋))
15	9, 13, 14	3syl 19	. . . . . . 7 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ (bday‘𝑋))
16	15	biantrud 541	. . . . . 6 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → ((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ (bday‘𝑋))))
17		ssint 4927	. . . . . . 7 ⊢ ((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})(bday‘𝑋) ⊆ 𝑧)
18		fvelimab 6983	. . . . . . . . . . . . . 14 ⊢ ((bday Fn No ∧ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No) → (𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} (bday‘𝑦) = 𝑧))
19	10, 11, 18	mp2an 705	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} (bday‘𝑦) = 𝑧)
20		sneq 4594	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
21	20	breq2d 5118	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
22	20	breq1d 5116	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
23	21, 22	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
24	23	rexrab 3657	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} (bday‘𝑦) = 𝑧 ↔ ∃𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧))
25	19, 24	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∃𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧))
26	25	imbi1i 352	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → (bday‘𝑋) ⊆ 𝑧) ↔ (∃𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧))
27		r19.23v 3193	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ No (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧) ↔ (∃𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧))
28		eqcom 2770	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑦) = 𝑧 ↔ 𝑧 = (bday‘𝑦))
29	28	anbi1ci 638	. . . . . . . . . . . . . 14 ⊢ (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) ↔ (𝑧 = (bday‘𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
30	29	imbi1i 352	. . . . . . . . . . . . 13 ⊢ ((((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧) ↔ ((𝑧 = (bday‘𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)) → (bday‘𝑋) ⊆ 𝑧))
31		impexp 456	. . . . . . . . . . . . 13 ⊢ (((𝑧 = (bday‘𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)) → (bday‘𝑋) ⊆ 𝑧) ↔ (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
32	30, 31	bitri 278	. . . . . . . . . . . 12 ⊢ ((((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧) ↔ (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
33	32	ralbii 3112	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ No (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ (bday‘𝑦) = 𝑧) → (bday‘𝑋) ⊆ 𝑧) ↔ ∀𝑦 ∈ No (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
34	26, 27, 33	3bitr2i 302	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → (bday‘𝑋) ⊆ 𝑧) ↔ ∀𝑦 ∈ No (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
35	34	albii 1852	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → (bday‘𝑋) ⊆ 𝑧) ↔ ∀𝑧∀𝑦 ∈ No (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
36		df-ral 3081	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})(bday‘𝑋) ⊆ 𝑧 ↔ ∀𝑧(𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) → (bday‘𝑋) ⊆ 𝑧))
37		ralcom4 3292	. . . . . . . . 9 ⊢ (∀𝑦 ∈ No ∀𝑧(𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)) ↔ ∀𝑧∀𝑦 ∈ No (𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
38	35, 36, 37	3bitr4i 306	. . . . . . . 8 ⊢ (∀𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})(bday‘𝑋) ⊆ 𝑧 ↔ ∀𝑦 ∈ No ∀𝑧(𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)))
39		fvex 6924	. . . . . . . . . 10 ⊢ (bday‘𝑦) ∈ V
40		sseq2 3939	. . . . . . . . . . 11 ⊢ (𝑧 = (bday‘𝑦) → ((bday‘𝑋) ⊆ 𝑧 ↔ (bday‘𝑋) ⊆ (bday‘𝑦)))
41	40	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑧 = (bday‘𝑦) → (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧) ↔ ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦))))
42	39, 41	ceqsalv 3493	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)) ↔ ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))
43	42	ralbii 3112	. . . . . . . 8 ⊢ (∀𝑦 ∈ No ∀𝑧(𝑧 = (bday‘𝑦) → ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ 𝑧)) ↔ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))
44	38, 43	bitri 278	. . . . . . 7 ⊢ (∀𝑧 ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})(bday‘𝑋) ⊆ 𝑧 ↔ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))
45	17, 44	bitri 278	. . . . . 6 ⊢ ((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))
46	16, 45	bitr3di 289	. . . . 5 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → (((bday‘𝑋) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ∧ ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ⊆ (bday‘𝑋)) ↔ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦))))
47	2, 46	bitrid 286	. . . 4 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → ((bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}) ↔ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦))))
48	47	pm5.32da 590	. . 3 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → (((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ (bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))))
49		df-3an 1105	. . 3 ⊢ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ (bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ (bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})))
50		df-3an 1105	. . 3 ⊢ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦))) ↔ ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅) ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦))))
51	48, 49, 50	3bitr4g 317	. 2 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → ((𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ (bday‘𝑋) = ∩ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))))
52	1, 51	bitrd 282	1 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → (bday‘𝑋) ⊆ (bday‘𝑦)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900  {csn 4588  ∩ cint 4910   class class class wbr 5106   “ cima 5659   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178

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-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-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-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179

This theorem is used by:  bday0b  28232  cuteq1  28236  oncutlt  28683