Theorem cardaleph 8383

Description: Given any transfinite cardinal number A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
cardaleph	|- ( ( om C_ A /\ ( card ` A ) = A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )

Distinct variable group:   x, A

Proof of Theorem cardaleph

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardon 8238	. . . . . . . . 9 |- ( card ` A ) e. On
2		eleq1 2454	. . . . . . . . 9 |- ( ( card ` A ) = A -> ( ( card ` A ) e. On <-> A e. On ) )
3	1, 2	mpbii 211	. . . . . . . 8 |- ( ( card ` A ) = A -> A e. On )
4		alephle 8382	. . . . . . . . 9 |- ( A e. On -> A C_ ( aleph ` A ) )
5		fveq2 5774	. . . . . . . . . . 11 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
6	5	sseq2d 3445	. . . . . . . . . 10 |- ( x = A -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` A ) ) )
7	6	rspcev 3135	. . . . . . . . 9 |- ( ( A e. On /\ A C_ ( aleph ` A ) ) -> E. x e. On A C_ ( aleph ` x ) )
8	4, 7	mpdan 666	. . . . . . . 8 |- ( A e. On -> E. x e. On A C_ ( aleph ` x ) )
9		nfcv 2544	. . . . . . . . . 10 |- F/_ x A
10		nfcv 2544	. . . . . . . . . . 11 |- F/_ x aleph
11		nfrab1 2963	. . . . . . . . . . . 12 |- F/_ x { x e. On | A C_ ( aleph ` x ) }
12	11	nfint 4209	. . . . . . . . . . 11 |- F/_ x |^| { x e. On | A C_ ( aleph ` x ) }
13	10, 12	nffv 5781	. . . . . . . . . 10 |- F/_ x ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
14	9, 13	nfss 3410	. . . . . . . . 9 |- F/ x A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
15		fveq2 5774	. . . . . . . . . 10 |- ( x = |^| { x e. On | A C_ ( aleph ` x ) } -> ( aleph ` x ) = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
16	15	sseq2d 3445	. . . . . . . . 9 |- ( x = |^| { x e. On | A C_ ( aleph ` x ) } -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
17	14, 16	onminsb 6533	. . . . . . . 8 |- ( E. x e. On A C_ ( aleph ` x ) -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
18	3, 8, 17	3syl 20	. . . . . . 7 |- ( ( card ` A ) = A -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
19	18	a1i 11	. . . . . 6 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( card ` A ) = A -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
20		fveq2 5774	. . . . . . . . 9 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` (/) ) )
21		aleph0 8360	. . . . . . . . 9 |- ( aleph ` (/) ) = om
22	20, 21	syl6eq 2439	. . . . . . . 8 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = om )
23	22	sseq1d 3444	. . . . . . 7 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A <-> om C_ A ) )
24	23	biimprd 223	. . . . . 6 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( om C_ A -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) )
25	19, 24	anim12d 561	. . . . 5 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( ( card ` A ) = A /\ om C_ A ) -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) ) )
26		eqss 3432	. . . . 5 |- ( A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) C_ A ) )
27	25, 26	syl6ibr 227	. . . 4 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( ( ( card ` A ) = A /\ om C_ A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
28	27	com12 31	. . 3 |- ( ( ( card ` A ) = A /\ om C_ A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
29	28	ancoms 451	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
30		vex 3037	. . . . . . . . . . . 12 |- y e. _V
31	30	sucid 4871	. . . . . . . . . . 11 |- y e. suc y
32		eleq2 2455	. . . . . . . . . . 11 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> ( y e. |^| { x e. On | A C_ ( aleph ` x ) } <-> y e. suc y ) )
33	31, 32	mpbiri 233	. . . . . . . . . 10 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> y e. |^| { x e. On | A C_ ( aleph ` x ) } )
34		fveq2 5774	. . . . . . . . . . . 12 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
35	34	sseq2d 3445	. . . . . . . . . . 11 |- ( x = y -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` y ) ) )
36	35	onnminsb 6538	. . . . . . . . . 10 |- ( y e. On -> ( y e. |^| { x e. On | A C_ ( aleph ` x ) } -> -. A C_ ( aleph ` y ) ) )
37	33, 36	syl5 32	. . . . . . . . 9 |- ( y e. On -> ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> -. A C_ ( aleph ` y ) ) )
38	37	imp 427	. . . . . . . 8 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> -. A C_ ( aleph ` y ) )
39	38	adantl 464	. . . . . . 7 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> -. A C_ ( aleph ` y ) )
40		fveq2 5774	. . . . . . . . . . 11 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` suc y ) )
41		alephsuc 8362	. . . . . . . . . . 11 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
42	40, 41	sylan9eqr 2445	. . . . . . . . . 10 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = (har ` ( aleph ` y ) ) )
43	42	eleq2d 2452	. . . . . . . . 9 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> A e. (har ` ( aleph ` y ) ) ) )
44	43	biimpd 207	. . . . . . . 8 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A e. (har ` ( aleph ` y ) ) ) )
45		elharval 7904	. . . . . . . . . 10 |- ( A e. (har ` ( aleph ` y ) ) <-> ( A e. On /\ A ~<_ ( aleph ` y ) ) )
46	45	simprbi 462	. . . . . . . . 9 |- ( A e. (har ` ( aleph ` y ) ) -> A ~<_ ( aleph ` y ) )
47		onenon 8243	. . . . . . . . . . . 12 |- ( A e. On -> A e. dom card )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> A e. dom card )
49		alephon 8363	. . . . . . . . . . . 12 |- ( aleph ` y ) e. On
50		onenon 8243	. . . . . . . . . . . 12 |- ( ( aleph ` y ) e. On -> ( aleph ` y ) e. dom card )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- ( aleph ` y ) e. dom card
52		carddom2 8271	. . . . . . . . . . 11 |- ( ( A e. dom card /\ ( aleph ` y ) e. dom card ) -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
53	48, 51, 52	sylancl 660	. . . . . . . . . 10 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
54		sseq1 3438	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A C_ ( card ` ( aleph ` y ) ) ) )
55		alephcard 8364	. . . . . . . . . . . 12 |- ( card ` ( aleph ` y ) ) = ( aleph ` y )
56	55	sseq2i 3442	. . . . . . . . . . 11 |- ( A C_ ( card ` ( aleph ` y ) ) <-> A C_ ( aleph ` y ) )
57	54, 56	syl6bb 261	. . . . . . . . . 10 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A C_ ( aleph ` y ) ) )
58	53, 57	bitr3d 255	. . . . . . . . 9 |- ( ( card ` A ) = A -> ( A ~<_ ( aleph ` y ) <-> A C_ ( aleph ` y ) ) )
59	46, 58	syl5ib 219	. . . . . . . 8 |- ( ( card ` A ) = A -> ( A e. (har ` ( aleph ` y ) ) -> A C_ ( aleph ` y ) ) )
60	44, 59	sylan9r 656	. . . . . . 7 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A C_ ( aleph ` y ) ) )
61	39, 60	mtod 177	. . . . . 6 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
62	61	rexlimdvaa 2875	. . . . 5 |- ( ( card ` A ) = A -> ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
63		onintrab2 6536	. . . . . . . . . . . . . 14 |- ( E. x e. On A C_ ( aleph ` x ) <-> |^| { x e. On | A C_ ( aleph ` x ) } e. On )
64	8, 63	sylib 196	. . . . . . . . . . . . 13 |- ( A e. On -> |^| { x e. On | A C_ ( aleph ` x ) } e. On )
65		onelon 4817	. . . . . . . . . . . . 13 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> y e. On )
66	64, 65	sylan 469	. . . . . . . . . . . 12 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> y e. On )
67	36	adantld 465	. . . . . . . . . . . 12 |- ( y e. On -> ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A C_ ( aleph ` y ) ) )
68	66, 67	mpcom 36	. . . . . . . . . . 11 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A C_ ( aleph ` y ) )
69	49	onelssi 4900	. . . . . . . . . . 11 |- ( A e. ( aleph ` y ) -> A C_ ( aleph ` y ) )
70	68, 69	nsyl 121	. . . . . . . . . 10 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` y ) )
71	70	nrexdv 2838	. . . . . . . . 9 |- ( A e. On -> -. E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
72	71	adantr 463	. . . . . . . 8 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
73		alephlim 8361	. . . . . . . . . . 11 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) )
74	64, 73	sylan 469	. . . . . . . . . 10 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) )
75	74	eleq2d 2452	. . . . . . . . 9 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> A e. U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) ) )
76		eliun 4248	. . . . . . . . 9 |- ( A e. U_ y e. |^| { x e. On | A C_ ( aleph ` x ) } ( aleph ` y ) <-> E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
77	75, 76	syl6bb 261	. . . . . . . 8 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) ) )
78	72, 77	mtbird 299	. . . . . . 7 |- ( ( A e. On /\ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
79	78	ex 432	. . . . . 6 |- ( A e. On -> ( Lim |^| { x e. On | A C_ ( aleph ` x ) } -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
80	3, 79	syl 16	. . . . 5 |- ( ( card ` A ) = A -> ( Lim |^| { x e. On | A C_ ( aleph ` x ) } -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
81	62, 80	jaod 378	. . . 4 |- ( ( card ` A ) = A -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
82	8, 17	syl 16	. . . . . 6 |- ( A e. On -> A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
83		alephon 8363	. . . . . . 7 |- ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) e. On
84		onsseleq 4833	. . . . . . 7 |- ( ( A e. On /\ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) e. On ) -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) ) )
85	83, 84	mpan2 669	. . . . . 6 |- ( A e. On -> ( A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) ) )
86	82, 85	mpbid 210	. . . . 5 |- ( A e. On -> ( A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) \/ A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
87	86	ord 375	. . . 4 |- ( A e. On -> ( -. A e. ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
88	3, 81, 87	sylsyld 56	. . 3 |- ( ( card ` A ) = A -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
89	88	adantl 464	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
90		eloni 4802	. . . . 5 |- ( |^| { x e. On | A C_ ( aleph ` x ) } e. On -> Ord |^| { x e. On | A C_ ( aleph ` x ) } )
91		ordzsl 6579	. . . . . 6 |- ( Ord |^| { x e. On | A C_ ( aleph ` x ) } <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) )
92		3orass 974	. . . . . 6 |- ( ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
93	91, 92	bitri 249	. . . . 5 |- ( Ord |^| { x e. On | A C_ ( aleph ` x ) } <-> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
94	90, 93	sylib 196	. . . 4 |- ( |^| { x e. On | A C_ ( aleph ` x ) } e. On -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
95	3, 64, 94	3syl 20	. . 3 |- ( ( card ` A ) = A -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
96	95	adantl 464	. 2 |- ( ( om C_ A /\ ( card ` A ) = A ) -> ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) \/ ( E. y e. On |^| { x e. On | A C_ ( aleph ` x ) } = suc y \/ Lim |^| { x e. On | A C_ ( aleph ` x ) } ) ) )
97	29, 89, 96	mpjaod 379	1 |- ( ( om C_ A /\ ( card ` A ) = A ) -> A = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   \/ w3o 970   = wceq 1399   e. wcel 1826   E.wrex 2733   {crab 2736   C_ wss 3389   (/)c0 3711   |^|cint 4199   U_ciun 4243   class class class wbr 4367   Ord word 4791   Oncon0 4792   Lim wlim 4793   suc csuc 4794   dom cdm 4913   ` cfv 5496   omcom 6599   ~<_ cdom 7433  harchar 7897   cardccrd 8229   alephcale 8230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-oi 7850  df-har 7899  df-card 8233  df-aleph 8234

This theorem is referenced by:  cardalephex  8384  tskcard  9070