Theorem eqcuts2 27782

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 27781	. 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 3949	. . . . 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 5110	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
5	3	breq1d 5108	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
6	4, 5	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
7	6	elrab3 3647	. . . . . . . . . 10 ⊢ (𝑋 ∈ No → (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
9	8	biimpar 477	. . . . . . . 8 ⊢ (((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) ∧ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)) → 𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)})
10		bdayfn 27745	. . . . . . . . 9 ⊢ bday Fn No
11		ssrab2 4032	. . . . . . . . 9 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12		fnfvima 7179	. . . . . . . . 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 1453	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday ‘𝑋) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14		intss1 4918	. . . . . . . 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 18	. . . . . . 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 531	. . . . . 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 4919	. . . . . . 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 6906	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
22	20	breq1d 5108	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
23	21, 22	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
24	23	rexrab 3654	. . . . . . . . . . . . 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 275	. . . . . . . . . . . 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 349	. . . . . . . . . . 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 3163	. . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑦) = 𝑧 ↔ 𝑧 = ( bday ‘𝑦))
29	28	anbi1ci 626	. . . . . . . . . . . . . 14 ⊢ (((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) ∧ ( bday ‘𝑦) = 𝑧) ↔ (𝑧 = ( bday ‘𝑦) ∧ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
30	29	imbi1i 349	. . . . . . . . . . . . 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 450	. . . . . . . . . . . . 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 275	. . . . . . . . . . . 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 3082	. . . . . . . . . . 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 299	. . . . . . . . . 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 1820	. . . . . . . . 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 3052	. . . . . . . . 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 3262	. . . . . . . . 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 303	. . . . . . . 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 6847	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
40		sseq2 3960	. . . . . . . . . . 11 ⊢ (𝑧 = ( bday ‘𝑦) → (( bday ‘𝑋) ⊆ 𝑧 ↔ ( bday ‘𝑋) ⊆ ( bday ‘𝑦)))
41	40	imbi2d 340	. . . . . . . . . 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 3480	. . . . . . . . 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 3082	. . . . . . . 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 275	. . . . . . 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 275	. . . . . 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 286	. . . . 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 283	. . . 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 579	. . 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 1088	. . 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 1088	. . 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 314	. 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 279	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)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399   ⊆ wss 3901  {csn 4580  ∩ cint 4902   class class class wbr 5098   “ cima 5627   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   No csur 27607   bday cbday 27609   <<s cslts 27753   |s ccuts 27755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-slts 27754  df-cuts 27756

This theorem is referenced by:  bday0b  27809  cuteq1  27813  oncutlt  28260