Theorem eqcuts2 28177

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 28176	. 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 3923	. . . . 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 4590	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	breq2d 5114	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
5	3	breq1d 5112	. . . . . . . . . . . 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 3645	. . . . . . . . . 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 28139	. . . . . . . . 9 ⊢ bday Fn No
11		ssrab2 4005	. . . . . . . . 9 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12		fnfvima 7239	. . . . . . . . 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 1480	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → (bday‘𝑋) ∈ (bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14		intss1 4922	. . . . . . . 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 4923	. . . . . . 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 6957	. . . . . . . . . . . . . 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 4590	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
21	20	breq2d 5114	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
22	20	breq1d 5112	. . . . . . . . . . . . . . 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 3653	. . . . . . . . . . . . 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 3189	. . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 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 3108	. . . . . . . . . . 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 1851	. . . . . . . . 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 3077	. . . . . . . . 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 3288	. . . . . . . . 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 6898	. . . . . . . . . 10 ⊢ (bday‘𝑦) ∈ V
40		sseq2 3934	. . . . . . . . . . 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 3489	. . . . . . . . 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 3108	. . . . . . . 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 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895  {csn 4584  ∩ cint 4906   class class class wbr 5102   “ cima 5653   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150

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-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-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-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151

This theorem is used by:  bday0b  28204  cuteq1  28208  oncutlt  28655