Theorem alephval2 10508

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 10010	. . . . 5 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
2	1	ralrimiv 3142	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3		alephon 10005	. . . 4 ⊢ (ℵ‘𝐴) ∈ On
4	2, 3	jctil 520	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5		breq2 5109	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
6	5	ralbidv 3174	. . . 4 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
7	6	elrab 3645	. . 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 9909	. . . . 5 ⊢ (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10		alephcard 10006	. . . . . 6 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
11	10	eqcomi 2745	. . . . 5 ⊢ (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
12	9, 11	eleq2s 2856	. . . 4 ⊢ (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13		omex 9579	. . . . . 6 ⊢ ω ∈ V
14		vex 3449	. . . . . 6 ⊢ 𝑧 ∈ V
15		entri3 10495	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧 ∨ 𝑧 ≼ ω))
16	13, 14, 15	mp2an 690	. . . . 5 ⊢ (ω ≼ 𝑧 ∨ 𝑧 ≼ ω)
17		carddom 10490	. . . . . . . . . 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 9922	. . . . . . . . . 10 ⊢ (card‘ω) = ω
20	19	sseq1i 3972	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
21	18, 20	bitr3i 276	. . . . . . . 8 ⊢ (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22		cardidm 9895	. . . . . . . . . 10 ⊢ (card‘(card‘𝑧)) = (card‘𝑧)
23		cardalephex 10026	. . . . . . . . . 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 10011	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
26	25	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
27		breq1 5108	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
28	14	cardid 10483	. . . . . . . . . . . . . 14 ⊢ (card‘𝑧) ≈ 𝑧
29		sdomen1 9065	. . . . . . . . . . . . . 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 510	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 ↔ 𝑧 ≺ (ℵ‘𝐴)))
33		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
34	33	breq1d 5115	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
35	34	rspcv 3577	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36		sdomirr 9058	. . . . . . . . . . . . . . 15 ⊢ ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37		sdomen2 9066	. . . . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . 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 3154	. . . . . . . . 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 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50		ne0i 4294	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
51		onelon 6342	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
52		alephgeom 10018	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53		alephon 10005	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℵ‘𝑦) ∈ On
54		ssdomg 8940	. . . . . . . . . . . . . . . . . . 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 8947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
58	56, 57	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59		domnsym 9043	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
60	58, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
61	51, 60	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦 ∈ 𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
62	61	expr 457	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
63	62	ralrimiv 3142	. . . . . . . . . . . 12 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64		r19.2z 4452	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
65	64	ex 413	. . . . . . . . . . . 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 3103	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)
68	66, 67	syl6ib 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 454	. . . . . . . 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 5109	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
76	75	ralbidv 3174	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
77	76	elrab 3645	. . . . . 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 3142	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
82		ssrab2 4037	. . 3 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83		oneqmini 6369	. . 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ∩ cint 4907   class class class wbr 5105  Oncon0 6317  ‘cfv 6496  ωcom 7802   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  cardccrd 9871  ℵcale 9872

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-har 9493  df-card 9875  df-aleph 9876  df-ac 10052

This theorem is referenced by: (None)