Theorem alephval2 9273

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 8780	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
2	1	ralrimiv 2948	. . . . 5 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3		alephon 8775	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
4	2, 3	jctil 558	. . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5		breq2 4587	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
6	5	ralbidv 2969	. . . . 5 ⊢ (𝑥 = (ℵ‘𝐴) → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
7	6	elrab 3331	. . . 4 ⊢ ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
8	4, 7	sylibr 223	. . 3 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
9	8	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
10		cardsdomelir 8682	. . . . 5 ⊢ (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
11		alephcard 8776	. . . . . 6 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
12	11	eqcomi 2619	. . . . 5 ⊢ (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
13	10, 12	eleq2s 2706	. . . 4 ⊢ (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
14		omex 8423	. . . . . 6 ⊢ ω ∈ V
15		vex 3176	. . . . . 6 ⊢ 𝑧 ∈ V
16		entri3 9260	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧 ∨ 𝑧 ≼ ω))
17	14, 15, 16	mp2an 704	. . . . 5 ⊢ (ω ≼ 𝑧 ∨ 𝑧 ≼ ω)
18		carddom 9255	. . . . . . . . . 10 ⊢ ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
19	14, 15, 18	mp2an 704	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
20		cardom 8695	. . . . . . . . . 10 ⊢ (card‘ω) = ω
21	20	sseq1i 3592	. . . . . . . . 9 ⊢ ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
22	19, 21	bitr3i 265	. . . . . . . 8 ⊢ (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
23		cardidm 8668	. . . . . . . . . 10 ⊢ (card‘(card‘𝑧)) = (card‘𝑧)
24		cardalephex 8796	. . . . . . . . . 10 ⊢ (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
25	23, 24	mpbii 222	. . . . . . . . 9 ⊢ (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
26		alephord 8781	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
27	26	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
28	15	cardid 9248	. . . . . . . . . . . . . . 15 ⊢ (card‘𝑧) ≈ 𝑧
29		sdomen1 7989	. . . . . . . . . . . . . . 15 ⊢ ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
30	28, 29	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
31		breq1 4586	. . . . . . . . . . . . . 14 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
32	30, 31	syl5rbbr 274	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
33	27, 32	sylan9bb 732	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 ↔ 𝑧 ≺ (ℵ‘𝐴)))
34		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
35	34	breq1d 4593	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
36	35	rspcv 3278	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
37		sdomirr 7982	. . . . . . . . . . . . . . . 16 ⊢ ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
38		sdomen2 7990	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
39	28, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
40		breq2 4587	. . . . . . . . . . . . . . . . 17 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
41	39, 40	syl5bbr 273	. . . . . . . . . . . . . . . 16 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
42	37, 41	mtbiri 316	. . . . . . . . . . . . . . 15 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
43	36, 42	nsyli 154	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
44	43	com12 32	. . . . . . . . . . . . 13 ⊢ ((card‘𝑧) = (ℵ‘𝑥) → (𝑥 ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
45	44	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥 ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
46	33, 45	sylbird 249	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
47	46	exp31 628	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))))
48	47	rexlimdv 3012	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
49	25, 48	syl5 33	. . . . . . . 8 ⊢ (𝐴 ∈ On → (ω ⊆ (card‘𝑧) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50	22, 49	syl5bi 231	. . . . . . 7 ⊢ (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
51	50	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
52		ne0i 3880	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅)
53		onelon 5665	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
54		alephgeom 8788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
55		alephon 8775	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℵ‘𝑦) ∈ On
56		ssdomg 7887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
58	54, 57	sylbi 206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
59		domtr 7895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
60	58, 59	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
61		domnsym 7971	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
62	60, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
63	53, 62	sylan2 490	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦 ∈ 𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
64	63	expr 641	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
65	64	ralrimiv 2948	. . . . . . . . . . . 12 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
66		r19.2z 4012	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
67	66	ex 449	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
68	52, 65, 67	syl2im 39	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
69		rexnal 2978	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)
70	68, 69	syl6ib 240	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
71	70	com12 32	. . . . . . . . 9 ⊢ ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
72	71	expimpd 627	. . . . . . . 8 ⊢ (𝑧 ≼ ω → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
73	72	a1d 25	. . . . . . 7 ⊢ (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
74	73	com3r 85	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
75	51, 74	jaod 394	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((ω ≼ 𝑧 ∨ 𝑧 ≼ ω) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)))
76	17, 75	mpi 20	. . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
77		breq2 4587	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
78	77	ralbidv 2969	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
79	78	elrab 3331	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧))
80	79	simprbi 479	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧)
81	80	con3i 149	. . . 4 ⊢ (¬ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑧 → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
82	13, 76, 81	syl56 35	. . 3 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}))
83	82	ralrimiv 2948	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
84		ssrab2 3650	. . 3 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
85		oneqmini 5693	. . 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}))
86	84, 85	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})
87	9, 83, 86	syl2anc 691	1 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   class class class wbr 4583  Oncon0 5640  ‘cfv 5804  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644  ℵcale 8645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-ac 8822

This theorem is referenced by: (None)