Theorem cardaleph 8459

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 8314	. . . . . . . . 9 |- ( card ` A ) e. On
2		eleq1 2532	. . . . . . . . 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 8458	. . . . . . . . 9 |- ( A e. On -> A C_ ( aleph ` A ) )
5		fveq2 5857	. . . . . . . . . . 11 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
6	5	sseq2d 3525	. . . . . . . . . 10 |- ( x = A -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` A ) ) )
7	6	rspcev 3207	. . . . . . . . 9 |- ( ( A e. On /\ A C_ ( aleph ` A ) ) -> E. x e. On A C_ ( aleph ` x ) )
8	4, 7	mpdan 668	. . . . . . . 8 |- ( A e. On -> E. x e. On A C_ ( aleph ` x ) )
9		nfcv 2622	. . . . . . . . . 10 |- F/_ x A
10		nfcv 2622	. . . . . . . . . . 11 |- F/_ x aleph
11		nfrab1 3035	. . . . . . . . . . . 12 |- F/_ x { x e. On | A C_ ( aleph ` x ) }
12	11	nfint 4285	. . . . . . . . . . 11 |- F/_ x |^| { x e. On | A C_ ( aleph ` x ) }
13	10, 12	nffv 5864	. . . . . . . . . 10 |- F/_ x ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
14	9, 13	nfss 3490	. . . . . . . . 9 |- F/ x A C_ ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } )
15		fveq2 5857	. . . . . . . . . 10 |- ( x = |^| { x e. On | A C_ ( aleph ` x ) } -> ( aleph ` x ) = ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) )
16	15	sseq2d 3525	. . . . . . . . 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 6605	. . . . . . . 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 5857	. . . . . . . . 9 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` (/) ) )
21		aleph0 8436	. . . . . . . . 9 |- ( aleph ` (/) ) = om
22	20, 21	syl6eq 2517	. . . . . . . 8 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = (/) -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = om )
23	22	sseq1d 3524	. . . . . . 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 563	. . . . 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 3512	. . . . 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 453	. 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 3109	. . . . . . . . . . . 12 |- y e. _V
31	30	sucid 4950	. . . . . . . . . . 11 |- y e. suc y
32		eleq2 2533	. . . . . . . . . . 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 5857	. . . . . . . . . . . 12 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
35	34	sseq2d 3525	. . . . . . . . . . 11 |- ( x = y -> ( A C_ ( aleph ` x ) <-> A C_ ( aleph ` y ) ) )
36	35	onnminsb 6610	. . . . . . . . . 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 429	. . . . . . . 8 |- ( ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) -> -. A C_ ( aleph ` y ) )
39	38	adantl 466	. . . . . . 7 |- ( ( ( card ` A ) = A /\ ( y e. On /\ |^| { x e. On | A C_ ( aleph ` x ) } = suc y ) ) -> -. A C_ ( aleph ` y ) )
40		fveq2 5857	. . . . . . . . . . 11 |- ( |^| { x e. On | A C_ ( aleph ` x ) } = suc y -> ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) = ( aleph ` suc y ) )
41		alephsuc 8438	. . . . . . . . . . 11 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
42	40, 41	sylan9eqr 2523	. . . . . . . . . 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 2530	. . . . . . . . 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 7978	. . . . . . . . . 10 |- ( A e. (har ` ( aleph ` y ) ) <-> ( A e. On /\ A ~<_ ( aleph ` y ) ) )
46	45	simprbi 464	. . . . . . . . 9 |- ( A e. (har ` ( aleph ` y ) ) -> A ~<_ ( aleph ` y ) )
47		onenon 8319	. . . . . . . . . . . 12 |- ( A e. On -> A e. dom card )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> A e. dom card )
49		alephon 8439	. . . . . . . . . . . 12 |- ( aleph ` y ) e. On
50		onenon 8319	. . . . . . . . . . . 12 |- ( ( aleph ` y ) e. On -> ( aleph ` y ) e. dom card )
51	49, 50	ax-mp 5	. . . . . . . . . . 11 |- ( aleph ` y ) e. dom card
52		carddom2 8347	. . . . . . . . . . 11 |- ( ( A e. dom card /\ ( aleph ` y ) e. dom card ) -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
53	48, 51, 52	sylancl 662	. . . . . . . . . 10 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A ~<_ ( aleph ` y ) ) )
54		sseq1 3518	. . . . . . . . . . 11 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ ( card ` ( aleph ` y ) ) <-> A C_ ( card ` ( aleph ` y ) ) ) )
55		alephcard 8440	. . . . . . . . . . . 12 |- ( card ` ( aleph ` y ) ) = ( aleph ` y )
56	55	sseq2i 3522	. . . . . . . . . . 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 658	. . . . . . 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 2949	. . . . 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 6608	. . . . . . . . . . . . . 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 4896	. . . . . . . . . . . . 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 471	. . . . . . . . . . . 12 |- ( ( A e. On /\ y e. |^| { x e. On | A C_ ( aleph ` x ) } ) -> y e. On )
67	36	adantld 467	. . . . . . . . . . . 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 4979	. . . . . . . . . . 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 2913	. . . . . . . . 9 |- ( A e. On -> -. E. y e. |^| { x e. On | A C_ ( aleph ` x ) } A e. ( aleph ` y ) )
72	71	adantr 465	. . . . . . . 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 8437	. . . . . . . . . . 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 471	. . . . . . . . . 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 2530	. . . . . . . . 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 4323	. . . . . . . . 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 301	. . . . . . 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 434	. . . . . 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 380	. . . 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 8439	. . . . . . 7 |- ( aleph ` |^| { x e. On | A C_ ( aleph ` x ) } ) e. On
84		onsseleq 4912	. . . . . . 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 671	. . . . . 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 377	. . . 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 466	. 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 4881	. . . . 5 |- ( |^| { x e. On | A C_ ( aleph ` x ) } e. On -> Ord |^| { x e. On | A C_ ( aleph ` x ) } )
91		ordzsl 6651	. . . . . 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 971	. . . . . 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 466	. 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 381	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 368   /\ wa 369   \/ w3o 967   = wceq 1374   e. wcel 1762   E.wrex 2808   {crab 2811   C_ wss 3469   (/)c0 3778   |^|cint 4275   U_ciun 4318   class class class wbr 4440   Ord word 4870   Oncon0 4871   Lim wlim 4872   suc csuc 4873   dom cdm 4992   ` cfv 5579   omcom 6671   ~<_ cdom 7504  harchar 7971   cardccrd 8305   alephcale 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-har 7973  df-card 8309  df-aleph 8310

This theorem is referenced by:  cardalephex  8460  tskcard  9148