Theorem eqscut2 27869

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

Assertion

Ref	Expression
eqscut2	⊢ ((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 eqscut2

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

Step	Hyp	Ref	Expression
1		eqscut 27868	. 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 4024	. . . . 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 4658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
4	3	breq2d 5178	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑋}))
5	3	breq1d 5176	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ({𝑥} <<s 𝑅 ↔ {𝑋} <<s 𝑅))
6	4, 5	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅)))
7	6	elrab3 3709	. . . . . . . . . 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 27836	. . . . . . . . 9 ⊢ bday Fn No
11		ssrab2 4103	. . . . . . . . 9 ⊢ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} ⊆ No
12		fnfvima 7270	. . . . . . . . 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 1451	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)} → ( bday ‘𝑋) ∈ ( bday “ {𝑥 ∈ No ∣ (𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅)}))
14		intss1 4987	. . . . . . . 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 4988	. . . . . . 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 6994	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 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 4658	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
21	20	breq2d 5178	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐿 <<s {𝑥} ↔ 𝐿 <<s {𝑦}))
22	20	breq1d 5176	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝑅 ↔ {𝑦} <<s 𝑅))
23	21, 22	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐿 <<s {𝑥} ∧ {𝑥} <<s 𝑅) ↔ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)))
24	23	rexrab 3718	. . . . . . . . . . . . 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 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 2747	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑦) = 𝑧 ↔ 𝑧 = ( bday ‘𝑦))
29	28	anbi1ci 625	. . . . . . . . . . . . . 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 3099	. . . . . . . . . . 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 1817	. . . . . . . . 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 3068	. . . . . . . . 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 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 6933	. . . . . . . . . 10 ⊢ ( bday ‘𝑦) ∈ V
40		sseq2 4035	. . . . . . . . . . 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 3529	. . . . . . . . 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 3099	. . . . . . . 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 578	. . 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 1089	. . 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 1089	. . 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 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976  {csn 4648  ∩ cint 4970   class class class wbr 5166   “ cima 5703   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   No csur 27702   bday cbday 27704   <<s csslt 27843   |s cscut 27845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846

This theorem is referenced by:  bday0b  27893  cuteq1  27896