Theorem alephval2 10497

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 9998	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
2	1	ralrimiv 3129	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3		alephon 9993	. . . 4 ⊢ (ℵ‘𝐴) ∈ On
4	2, 3	jctil 519	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5		breq2 5104	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
6	5	ralbidv 3161	. . . 4 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
7	6	elrab 3648	. . 3 ⊢ ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
8	4, 7	sylibr 234	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
9		cardsdomelir 9899	. . . . 5 ⊢ (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10		alephcard 9994	. . . . . 6 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
11	10	eqcomi 2746	. . . . 5 ⊢ (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
12	9, 11	eleq2s 2855	. . . 4 ⊢ (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13		omex 9566	. . . . . 6 ⊢ ω ∈ V
14		vex 3446	. . . . . 6 ⊢ 𝑧 ∈ V
15		entri3 10483	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧 ∨ 𝑧 ≼ ω))
16	13, 14, 15	mp2an 693	. . . . 5 ⊢ (ω ≼ 𝑧 ∨ 𝑧 ≼ ω)
17		carddom 10478	. . . . . . . . . 10 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
18	13, 14, 17	mp2an 693	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
19		cardom 9912	. . . . . . . . . 10 ⊢ (card‘ω) = ω
20	19	sseq1i 3964	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
21	18, 20	bitr3i 277	. . . . . . . 8 ⊢ (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22		cardidm 9885	. . . . . . . . . 10 ⊢ (card‘(card‘𝑧)) = (card‘𝑧)
23		cardalephex 10014	. . . . . . . . . 10 ⊢ (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
24	22, 23	mpbii 233	. . . . . . . . 9 ⊢ (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
25		alephord 9999	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
26	25	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
27		breq1 5103	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
28	14	cardid 10471	. . . . . . . . . . . . . 14 ⊢ (card‘𝑧) ≈ 𝑧
29		sdomen1 9063	. . . . . . . . . . . . . 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 286	. . . . . . . . . . . 12 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
32	26, 31	sylan9bb 509	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 ↔ 𝑧 ≺ (ℵ‘𝐴)))
33		fveq2 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
34	33	breq1d 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
35	34	rspcv 3574	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36		sdomirr 9056	. . . . . . . . . . . . . . 15 ⊢ ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37		sdomen2 9064	. . . . . . . . . . . . . . . . 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 5104	. . . . . . . . . . . . . . . 16 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
40	38, 39	bitr3id 285	. . . . . . . . . . . . . . 15 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
41	36, 40	mtbiri 327	. . . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
45	32, 44	sylbird 260	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
46	45	rexlimdva2 3141	. . . . . . . . 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 242	. . . . . . 7 ⊢ (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
49	48	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50		ne0i 4295	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
51		onelon 6352	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
52		alephgeom 10006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53		alephon 9993	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℵ‘𝑦) ∈ On
54		ssdomg 8951	. . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
57		domtr 8958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
58	56, 57	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59		domnsym 9045	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
61	51, 60	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦 ∈ 𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
62	61	expr 456	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
63	62	ralrimiv 3129	. . . . . . . . . . . 12 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64		r19.2z 4454	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
65	64	ex 412	. . . . . . . . . . . 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 251	. . . . . . . . . 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 453	. . . . . . . 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 860	. . . . 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 5104	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
76	75	ralbidv 3161	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
77	76	elrab 3648	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
78	77	simprbi 497	. . . . 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 3129	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
82		ssrab2 4034	. . 3 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83		oneqmini 6380	. . 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 686	1 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ∩ cint 4904   class class class wbr 5100  Oncon0 6327  ‘cfv 6502  ωcom 7820   ≈ cen 8894   ≼ cdom 8895   ≺ csdm 8896  cardccrd 9861  ℵcale 9862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-ac2 10387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-oi 9429  df-har 9476  df-card 9865  df-aleph 9866  df-ac 10040

This theorem is referenced by: (None)