Theorem alephval2 10615

Description: An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)

Assertion

Ref	Expression
alephval2	⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem alephval2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephordi 10117	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
2	1	ralrimiv 3135	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3		alephon 10112	. . . 4 ⊢ (ℵ‘𝐴) ∈ On
4	2, 3	jctil 518	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5		breq2 5157	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
6	5	ralbidv 3168	. . . 4 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
7	6	elrab 3681	. . 3 ⊢ ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
8	4, 7	sylibr 233	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
9		cardsdomelir 10016	. . . . 5 ⊢ (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10		alephcard 10113	. . . . . 6 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
11	10	eqcomi 2735	. . . . 5 ⊢ (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
12	9, 11	eleq2s 2844	. . . 4 ⊢ (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13		omex 9686	. . . . . 6 ⊢ ω ∈ V
14		vex 3466	. . . . . 6 ⊢ 𝑧 ∈ V
15		entri3 10602	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧 ∨ 𝑧 ≼ ω))
16	13, 14, 15	mp2an 690	. . . . 5 ⊢ (ω ≼ 𝑧 ∨ 𝑧 ≼ ω)
17		carddom 10597	. . . . . . . . . 10 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
18	13, 14, 17	mp2an 690	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
19		cardom 10029	. . . . . . . . . 10 ⊢ (card‘ω) = ω
20	19	sseq1i 4008	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
21	18, 20	bitr3i 276	. . . . . . . 8 ⊢ (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22		cardidm 10002	. . . . . . . . . 10 ⊢ (card‘(card‘𝑧)) = (card‘𝑧)
23		cardalephex 10133	. . . . . . . . . 10 ⊢ (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
24	22, 23	mpbii 232	. . . . . . . . 9 ⊢ (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
25		alephord 10118	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
26	25	ancoms 457	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
27		breq1 5156	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
28	14	cardid 10590	. . . . . . . . . . . . . 14 ⊢ (card‘𝑧) ≈ 𝑧
29		sdomen1 9159	. . . . . . . . . . . . . 14 ⊢ ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
30	28, 29	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
31	27, 30	bitr3di 285	. . . . . . . . . . . 12 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
32	26, 31	sylan9bb 508	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 ↔ 𝑧 ≺ (ℵ‘𝐴)))
33		fveq2 6901	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
34	33	breq1d 5163	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
35	34	rspcv 3604	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36		sdomirr 9152	. . . . . . . . . . . . . . 15 ⊢ ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37		sdomen2 9160	. . . . . . . . . . . . . . . . 17 ⊢ ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
38	28, 37	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
39		breq2 5157	. . . . . . . . . . . . . . . 16 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
40	38, 39	bitr3id 284	. . . . . . . . . . . . . . 15 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
41	36, 40	mtbiri 326	. . . . . . . . . . . . . 14 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
42	35, 41	nsyli 157	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
43	42	com12 32	. . . . . . . . . . . 12 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → (𝑥 ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
44	43	adantl 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
45	32, 44	sylbird 259	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
46	45	rexlimdva2 3147	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
47	24, 46	syl5 34	. . . . . . . 8 ⊢ (𝐴 ∈ On → (ω ⊆ (card‘𝑧) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
48	21, 47	biimtrid 241	. . . . . . 7 ⊢ (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
49	48	adantr 479	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50		ne0i 4337	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
51		onelon 6401	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
52		alephgeom 10125	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53		alephon 10112	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℵ‘𝑦) ∈ On
54		ssdomg 9031	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
56	52, 55	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
57		domtr 9038	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
58	56, 57	sylan2 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59		domnsym 9137	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
61	51, 60	sylan2 591	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦 ∈ 𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
62	61	expr 455	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
63	62	ralrimiv 3135	. . . . . . . . . . . 12 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64		r19.2z 4499	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
65	64	ex 411	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
66	50, 63, 65	syl2im 40	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
67		rexnal 3090	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)
68	66, 67	imbitrdi 250	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
69	68	com12 32	. . . . . . . . 9 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
70	69	expimpd 452	. . . . . . . 8 ⊢ (𝑧 ≼ ω → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
71	70	a1d 25	. . . . . . 7 ⊢ (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
72	71	com3r 87	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
73	49, 72	jaod 857	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((ω ≼ 𝑧 ∨ 𝑧 ≼ ω) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
74	16, 73	mpi 20	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
75		breq2 5157	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
76	75	ralbidv 3168	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
77	76	elrab 3681	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
78	77	simprbi 495	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)
79	78	con3i 154	. . . 4 ⊢ (¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
80	12, 74, 79	syl56 36	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}))
81	80	ralrimiv 3135	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
82		ssrab2 4076	. . 3 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83		oneqmini 6428	. . 3 ⊢ ({𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On → (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}))
84	82, 83	ax-mp 5	. 2 ⊢ (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
85	8, 81, 84	syl2an2r 683	1 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3419  Vcvv 3462   ⊆ wss 3947  ∅c0 4325  ∩ cint 4954   class class class wbr 5153  Oncon0 6376  ‘cfv 6554  ωcom 7876   ≈ cen 8971   ≼ cdom 8972   ≺ csdm 8973  cardccrd 9978  ℵcale 9979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-ac2 10506

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9553  df-har 9600  df-card 9982  df-aleph 9983  df-ac 10159

This theorem is referenced by: (None)